MECHANICAL DRAWING. 



PROGRESSIVE EXERCISES 



AND 



PRACTICAL HINTS. 



FOR THE USE OF ALL WHO WISH TO ACQUIRE T. 

OR WITHOUT THE AiD OF AN INSTRUCTOR. 




CHARLES WILLIAM ^IacCORD, A.M., Sc.D., 

PROFESSOR OF MECHANICAL DRAWING IN THE STEVENS INSTITUTE OF TECHNOLOGY, HOBOKEN, N.J. 

FORMERLY CHIEF DRAUGHTSMAN FOR CAPT. JOHN ERICSSON; 

AUTHOR OF " KINEM.ATICS," " A PRACTICAL TREATISE ON THE SLIDE-VALVE AND ECCENTRIC," 

AND VARIOUS MONOGRAPHS ON MECHANISM. 



, c.p 19 1892 , 



%TtO(o X ^ 



NEW YORK : 

JOHN WILEY & SONS, 

53 East Tenth Street. 

1892. 



Copyright, 1892, 
By CHARLES WILLIAM MacCORD. 



Robert Ubdmmond, Fekbis Bros,, 

Electrotyper. Printers, 

*" and 446 Pearl St., 32j pearl Street. 
New York jlew York. 



{ 






PREFACE TO PART I. 




It is proper to state that the second part of this book was separately pubHshed some 
time before the first part was written ; which accounts for the fact that the chapter on Draw- 
ing Instruments is found at the end, and not at the beginning, where it might naturahy be 
looked for. 

The exercises contained in Part I embrace the essential features of a course of instruction 
which has been successfully pursued for tiventy years. Those given in the first two chapters 
are intended not only to train the eye and the hand in the use of instruments, but also to 
form the habit of exercising forethought, judgment, and taste in relation to the important 
matter of arrangement. 

Reasonable skill in e.xecution having been acquired, the next step is to the delineation of 
solid objects. And in the treatment of projections the aim is to lead the student to draw 
these in a matter-of-fact way, as they would appear from different points of view ; the express 
design being to keep out of sight as far as possible the artificial and often useless stage 
machinery of Descriptive Geometry. 

In this manner the beginner is enabled almost at once to construct intelligently simple 
detail drawings that have a practical meaning and a use, which of itself is a great incentive to 
further efforts. But this is not the only nor the strongest reason for adopting this course. 
There are many to whom the representations of tangible and actual things are perfectly clear, 
while those of abstractions, such as points, lines, or planes, in space, are obscure if net unin- 
telligible ; nevertheless they may and often do become expert and valuable practical draughts- 
men. 

But there are none who will not find a thorough familiarity with the former to be of the 
greatest advantage in the study of the latter,^one does not study the structure of language 
before he begins to use it, — and in this subject, as in many others, some knowledge of the 
concrete is essential to a ready apprehension of the abstract. 

C. W. MacCord. 

HoBOKEN, N. J., July 26, 1892. 



CONTENTS, 



CHAPTER I. 

PAGE. 

Elementary Exercises: Straight Lines and Circles, i 

CHAPTER n. 
Exercises in the Drawing of Non-Circular Curves 26 



CHAPTER HI. 
The Principles of Projection, 53 

CHAPTER IV. 
Objects in Inclined Positions 63 

CHAPTER V. 

Of the Helix, and its Application in the Drawing of Screws, 79 

CHAPTER VI. 
Intersections and Developments of Surfaces, 90 

CHAPTER VII. 

ISOMETRICAL DRAWINGS, CaVALIER PROJECTIONS, AND PSEUDO-PERSPECTIVE, . . . . II7 

CHAPTER VIII. 

The Spur-Wheel and the Bevel-Wheel in Inclined Positions. Construction of the 
Close Fitting Worm and Wheel. Construction of the Screw-Propeller. Stand- 
ard Sectioning, Etc i-, 

INDEX 147 

V 



MECHANICAL DRAWING. 



CHAPTER I. 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 

1. Mechanical drawing and geometrical drawing are by some writers used as synony- 
mous terms : in many treatises tlie subject is opened up by tlie explanation of various geo- 
metrical problems, and the first exercises consist in drawing, according to. the explanation, 
diagrams representing their solution — thus combining elementary instruction in the science of 
geometry with that in the art of drawing. 

No doubt a knowledge of that science is essential to proficiency in the art ; but it is 
not clear that to study both at once is the best way to acquire either. Those to whom the 
following pages are addressed, then, are presumed to have gained from other sources an 
acquaintance with such geometrical principles as may be made use of in the course of 
the work. 

2. A single illustration will serve to show that a distinction may justly be made between 
the terms above specified. 

Let it be required in Fig. i to draw through Pa perpendicular to AB. 



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Fig. I. 

The geometrical process is as follows : 

1. Describe about P with any assumed radius a circular arc cutting AB in the points C 
andZ*. 

2. Describe about C and D, with equal radii assumed at pleasure, two other arcs intersect- 
ing in E. 

3. Draw the straight line PE\ it is the perpendicular required. 



2 MECHANICAL DRAWING. 

Now if in the above or any other case, the required result is attained by actually executing the 
geometrical processes, the operation may properly be called geometrical drawing. And it is 
to be noted that the only instruments admissible are the dividers or compasses, and the 
ruler or straight-edge. The methods are precise, but the manipulations are often tedious. 

3. But by adding a few other simple implements, many operations of constant occurrence 
in making industrial drawings which involve geometrical principles can be performed with 
far greater facility, and yet with all the accuracy needed for practical purposes. In illustra- 
tion may be mentioned the drawing of horizontal and vertical lines by means of the T-square; 
the use of the scale for bisecting and subdividing lines; and the use of the triangles for draw- 
ing lines parallel or perpendicular to others in any position, for inscribing or circumscribing 
certain polygons, as well as for drawing tangents to circles and finding the points of contact. 
And that may appropriately be called mechanical drawing, in which the use of these and 
any other labor-saving appliances or devices is permitted. As a matter of fact, they are used 
in the execution of by far the greater part of the drawings made for constructive purposes. 

4. Accepting the latter definition, the following exercises are selected and arranged with 
a view to imparting facility and accuracy in the use of the instruments. Before proceeding 
to consider them, it may be well to give a few hints as to the use of the pencil; it is impor- 
tant to begin right, as habits once formed are not easily changed. 

1. Pencil lightly. Outlines to be inked in should, however, be distinct and firm ; centre 
lines and the like should be finer than these, while " construction lines," etc., which are not to 
be inked in, should be the finest of all. 

2. Pencil clearly. Keep the ruling pencils and the instrument leads sharp. Leave in 
no superfluous or " false" lines ; if aline is by any oversight drawn in erroneously, do not^ 
leave it to be rubbed out after inking in the rest, but erase it at once. 

3. Pencil in only what is to be inked in, whenever practicable ; it is easier to leave out 
lines, or parts of lines, than to rub them out : and the neatest draughtsman is the one who 
does the work with the least use of either pencil or rubber. Also, lines which are to be dotted 
in ink should be dotted in pencilling. 

4. Whenever it is possible, mark the extremities of a line by means of the scale, before 
drmving the line. This is often much facilitated by placing the scale against the edge of the 
T-square ; and it enables us to draw the line at once of its exact length, in compliance with 
the preceding suggestion. 

5. In setting of? measurements, and in subdividing lines, make use of the scale whenever 
it can be done, and use the dividers or compasses only when it is unavoidable. 

6. Do as much as possible with one setting or adjustment of any instrument, particularly 
with the dividers or compasses. Much time may be wasted by neglect of this simple precept. 

5. In short, the drawing should be made in pencil as nearly as possible as it is to be made 
in ink. It may take more time to execute the pencilling, but when the habit has been once 
formed, the difference is less than might be imagined, and is often nearly balanced by the 
saving of time in the inking in. A chief draughtsman may wish personally to lay out work, 
which he can, if thus executed, safely leave to be inked in by his assistants ; and again, a draw- 
ing thus made can upon occasion be traced at once, without being put in ink at all, by which 
a material saving of time mav be effected. 



ELEMENTARY EXERCISES : STRAIGHT LINES AND CIRCLES. 3 

Having mentioned the element of time, it may be well here to add a caution against 
undue haste, a frequent cause of oversights and errors, which is apt to lead to careless or slov- 
enly work. ^^ Make haste siozvly" is the golden rule of the draughtsman, whose most valuable 
qualification is accuracy; the next is neatness ; and the third in importance is speed, which 
is valuable only in so far as it is the result of skill and confidence gained by careful practice, 
— and worse than worthless if acquired at the expense of reliability. 

6. It is recommended for the sake of neatness and uniformity that the beginner should 
make a series of sheets measuring 19 inches by 12, a size requiring a quarter sheet of Double 
Elephant paper. A margin of one inch all around leaves a rectangle of 17 inches by 10, to be 
enclosed b}' a plain or ornamental border. Within this the drawings forming the exercises 
are to be arranged, with such symmetry as the subjects may admit ; and made of such 
dimensions that they shall neither be so small as to seem lost in the surrounding space, nor 
so large as to appear crowded. 

The effort to accomplish this gives from the outset a training in judgment and fore- 
thought which could not be hoped for were exercises of the same nature gone through with 
regardless of size or of external limitations. 

7. The sheet should be placed as near as may be to the lower left-hand corner of the 
drawing-board, and may be secured by means of four drawing-pins. But, tiuo pins will always 
suffice to keep the paper in place, and one or more must be temporarily removed if they 
interfere with tlie T-square or triangles, which should always We flat on the paper. This being 
arranged, the outer lines may be drawn from edge to edge of the paper, since the sheet when 
finished is to be cut to the size thus indicated. This final trimming should alwa)-s be done 
with shears ; and never with a knife, the use of which would injure the drawing-board, and 
sooner or later ruin the T-square, if, as is too likely, it were used as a ruler for this purpose. 

8. Fig. 2 represents a completed sheet before trimming ; and the exercises which it con- 
tains are such that by due attention to the preceding suggestions it is easy to finish up the 
whole without using the rubber at all. Thus, having set ofT a point one inch below the upper 
line, set the T-square against the left side of the drawing-board, and place the scale along its 
upper edge. Slide the T-square up until the upper edge of the scale passes through the 
point; then sliding the scale along the square, set off a point one inch inside of each of 
the outer vertical lines, thus locating the upper corners of the border. Repeating this at the 
lower side of the sheet, the border can be drawn of its exact size. Again, mark a point one 
inch below the upper line of the border, and using the T-square and the scale together as 
before, slide the latter along until its zero point is on the left-hand line of the border, and 
with the pencil mark the distances i, 5, 6h, loh, 12, and 16, zvitliout moving the scale. These 
points locate the upper corners of the three rectangular figures drawn in the upper part of 
the sheet; and it need not be described in detail how in a similar manner all the corners of all 
the rectangles, as well as the centre lines of the circles drawn in two of them, may be located, 
so that all the pencil-lines may be drawn without " overrunning " or intersecting. 

9. Particular attention is called to the manner of using the scale above mentioned, viz., 
the setting off of the sums of successive distances from a fixed zero. A very common error is 
to set off the first, then to move the scale, and placing the zero at the point thus marked, 
to set off the second, and so on. Since there is always a possibility of making an error in 



MECHANICAL DRA WING. 



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ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 5 

setting off any distance whatever, the result of this method is likely to be that the total 
distance is not exactly equal to the sum of its parts, because the errors may accumulate. 
In adopting the method recommended, there is at least the security that the total error in 
laying off any distance within the limit of the scale will not exceed that which may exist 
in the measurement of one of the parts. This is of greater importance in making working 
drawings, but a correct method should be used from the very beginning. 

10. In regard to "inking in," the following general instructions apply in all cases when 
the best results are desired : 

1. Do not take the pen in hand until the pencil is done with ; then ink in all the outlines, 
of uniform thickness, drawing the circles and circular arcs, if there are any, first, and leaving 
the straight lines till the last. 

2. The centre-lines and construction-lines, if there are any, are to be put in next ; and 
these should be " hair-lines," i.e., the finest of all upon the sheet. 

3. When this is done the sheet should be cleaned, and all traces of pencilling removed. 

4. The dotted "dimension-lines," if any, should now be put in, and the arrow-heads and 
figures next. 

5. If any parts are shown in section, the "hatching" or "sectioning," by which this is 
indicated, should be done next. 

6. If there be any " line-shading," by which the forms of uncut surfaces are indicated, 
this should be now executed. 

7. The " shadow-lines," which indicate the edges that intercept the light and cast 
shadows, are put in last of. all. 

Some terms not yet explained have necessarily been introduced in the above : these will 
be considered in due course. 

11. Thickness of Lines. — Lines are distinguished as fine or coarse, thin or thick, light 
or heavy, indifferently. The absolute thickness of line which is suitable in any given case 
cannot be specified by a positive rule, but must be learned by observation and experience. 
In a general way, however, it may be stated that the smaller the objects represented, and the 
smaller the scale upon which they are drawn, the finer the lines should be. But it is quite 
clear that the thickness cannot vary precisely in proportion to the scale, because drawings 
made on even the smallest scales must be defined by lines which are distinctly visible. 
Whether thick or thin, however, the lines should be smooth, even, and continuous, and if in 
ink, perfectly black ; as the finish of a mechanical drawing depends very largely upon the 
quality of the lines. 

12. Junction of Lines. — When two lines meet and terminate at the same point, as for 
example in the corners of the rectangles in Fig. 2, especial care must be taken to make the 
junction sharp and clean, as shown in the illustrations on the left, in Fig. 3. If through 
haste or carelessness one or both the lines be made too long or too short, as illustrated on 
the right, the effect is exceedingly bad, no matter how smooth the lines. Minute defects like 
this are the first to catch the eye of the expert, while more important errors might escape 
notice ; and they hold the eye, too, like little flaws in a mirror or little cracks in a vase. 
Indeed, the finish of a drawing depends much upon such small and apparently insignificant 
things ; and besides, it is a fair inference that one who is careless about these points of detail 



6 MECHANICAL DRAWING. 

may be so in regard to greater matters. These remarks apply with equal stringency to the 
plain "border-line" around Fig. 2, which of itself is, no less than the figures contained 






Fig. 3. 



within it, an exercise in mechanical drawing; it should be treated with equal deference, and 
drawn with the .same care. 

13. Sectioning. — When an object or any portion of it is supposed to be cut by a plane, 
as in mechanical structures is often necessary in order to show the interior arrangement, the 
surface of the cutting plane is conventionally indicated by "hatching," to which reference was 
made in (10). This consists merely in covering the surface with fine parallel lines, making 
ordinarily an angle of 45° with the bounding-lines of the surface when these are straight, and 
an angle of 45° with the horizontal line when the outlines are curved ; which inclinations, 
however, are subject to modifications in cases to be subsequently discussed. Various parts 
of the exercise in Fig. 2 are shown as thus " hatched," or, as it is more commonly called, 
" sectioned," which will serve to give practice in the use of the pen and also of the triangles, 
meantime training the eye also in judging of equal spaces ; for it is to be understood that 
the distances between the lines are not to be spaced off by any instrument, but determined by 
the eye alone. The operation consists simply in ruling one line after another, sliding the 
triangle along the edge of the T-square for the same distance after drawing each one. 

14. When a few lines have thus been drawn in succession, an unconscious rhythmic 
action, as it were, is established, just as one beats time to fast or slow music without thinking, 
and the manipulation becomes to a great extent mechanical. For this reason a drawing 
should never be sectioned in pencil first, but this should be done in ink at once : the pencilled 
sectioning may be excellent, but if inked in afterward, the result is likely to be as bad as if 
one were to write his name with a pencil and then try to go over the lines with a pen. And 
for the same reason the sectioning of a drawing should, whenever it is possible, be done at 
one continuous operation ; for if a part be done one day, and another the next, there is likely 
to be a perceptible difference in the tone of the two parts. And again, in making a tracing 
from a completed drawing no attempt should be made to trace the section-lining: the out- 
lines having been copied, a sheet of white paper is slipped under the tracing-cloth, and the 
sectioning is done as upon a new drawing. 



ELEMENTARY EXERCISES : STRAIGHT LINES AND CIRCLES. 7 

15. No absolute rule can be laid down as to the width of the spaces between the lines. 
The general effect to be aimed at is that of a light tint covering the surface ; hence it is clear 
that the lines of the sectioning should be finer than the outlines. If a narrow surface be 
sectioned with wide spaces, the effect will be coarse and unpleasing; per contra, the larger the 
surface, the wider the spaces may be made. If the spaces be too narrow, the effect will be that 
of a dark tint, tending to crush out the outlines: and besides, it not only requires more time to 
do the work, but any error in the spacing will be much more conspicuous ; and the evenness of 
the tone evidently depends upon the uniformity of the spaces. It is to be noted also that the 
section-lines must begin and end at just the right points, neither overrunning nor falling short 
of the outlines ; otherwise the effect is rough and ragged. 

16. But the evenness of the tone also depends upon the uniformity of the lines in both 
tint and thickness. If the ink thickens in the pen, the result is a line either too dark or too 
thick, or both. The so-called " prepared ink " is peculiarly liable to become viscid in the pen, 
thus producing this exasperating result ; and there is not a bottle of it, of any make, with 
which it would be safe to undertake the sectioning, much less the line-shading, of any elabo- 
rate plan. Throw the bottle out of the window; mix up a liberal quantity of good China 
ink with clear water in a clean saucer, a little thinner than for outlining ; test the pen ; practise 
a minute on some scrap of paper until hand and eye act in unison and the proper space is 
determined ; then proceed with confidence and hope of success. In Fig. 2 one of the rect- 
angles is ruled with equidistant horizontal lines, in the manner of line-shading a plane surface. 
These, too, are to be spaced by the eye, and the above remarks apply to them as well. 

17. Shadow-lines. — In Fig. 2 it will be observed that some of the outlines are much 
heavier than others ; and no one can fail to be struck by the effect of relief thus produced, 
nor to appreciate that the figures are not mere diagrams, but represent objects of sensible 
thickness, which if resting upon a plane surface would cast shadows upon it. The lines which 
are thus made heavy are technically called shadow-lines ; and in all cases they represent the 
sharp edges of surfaces, which intercept the light. Thus they not only give the power of 
producing relief in outline drawings, but afford at once the means of distinguishing an object 
which has sharp edges, as for instance a prism, from one which has not, such as a sphere or a 
cylinder. Of course it would be out of place here to attempt the full explanation of how to 
determine in all cases what lines do cast the shadows. But this must depend to a great 
extent upon the direction from which the light comes ; and for the sake of uniformity it has 
been agreed that on the drawing this direction shall in all cases appear to make an angle 
of 45° with the horizontal line, as indicated by the arrow in Fig. 4, it being understood that 
the rays of light are ViOt parallel ^o the paper, but come from a source above the object as 
well as to the left of it. 

18. Suppose then that in this figure ABCD represents a square block lying on the paper, 
and perforated by a square opening EFGH, all the edges being sharp. It will be seen at 
once that AB, BC, EH, and HG are the lines which cast shadows, and they are made heavy 
accordingly. Let X represent a torus, or ring, such as can be made by bending a cylindrical 
rod around a mandrel : this body would certainly cast a shadow, but it has no edges, and 
therefore no shadow-lines Let F represent an annulus, or ring formed by bending a square 
bar around a mandrel: then the outer and the inner circles represent edges, and portions of 



8 



MECHANICAL DRAWING. 



both will cast shadows, falling away from the light. Draw rays of light tangent to the outer 
circle at in, n : then, as any one can readily satisfy himself by laying a disk, say a coin, upon 
the table in the sunlight, the semicircumference vian will not cast a shadow, while Tidi/i will 
do so ; moreover, the shadow will be broadest opposite d, the point midway between ;« and 




Fig. 4. 

;/, gradually diminishing in breadth as it approaches these points of tangency, where it will 
disappear. And, similarly, the opposite half of the inner circumference will cast a shadow, 
likewise tapering out to nothing at the points of tangency to parallel rays. 

19. Of course it is not necessary actually to draw these rays, since, as indicated in the 
figure, the points of tangency can be at once marked by means of the 45° triangle. 

In order to draw this tapering shadow-line the compasses must be put in motion 
before the pen touches the paper at n ; the pressure is gradually increased as the pen approaches 
b, gradually diminished as it approaches m, and the pen is removed from the paper ivhile still 
in motion. The whole action is very similar to that of free-hand flourishing with a writing- 
pen, and is easily acquired with a little practice. 

If the line is to be quite heavy, it may be necessary to go twice or even three times over 
the broader part. Beginners are very apt to think that the object can be accomplished by 
slightly shifting the centre — a very dangerous expedient, unless the line is to be made exces- 
sively thick, as it might be for a lecture-room diagram ; for it is almost impossible to move 
the needle-point through a distance sufUciently minute without having it slip back again into 
the hole made by it in the original position. For all ordinary purposes the method above 
advised is the safest and the best. 

20. Border-lines. — The preceding instructions embrace all that is necessary for the 
completion of the exercises given in Fig. 2, and apply equally to all those following which 
come within their scope. A word may now be said as to the border-line by which they are 
surrounded. In general such an ornamental adjunct is not appropriate in connection with 
separate detail drawings ; but when a series of sheets of uniform size is to be made, it may 
be introduced with good effect. Around topographical drawings, however simple and crude, 
a border of some kind is regarded as almost indispensable. But wherever used, it is in a 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 



sense ornamental and superfluous, it challenges criticism, and should be executed with 
scrupulous care. Subsequently some examples will be given of more ornate borders with 
correspondingly elaborate "corner-pieces," suitable for drawings of a pictorial nature; for 
the present, the less pretentious " plain border," consisting, as shown in Fig. 2, of a fine 
interior line and a thicker one without, will suffice to exercise the skill of the tyro. 

The absolute thickness of the outer line is to a great extent dependent upon individual 
taste ; the standard adopted in topographical work is y^^ of the breadth of the map — which 
would give, for the borders here proposed, enclosing a space lo inches wide, an outside line 
yig- of an inch thick : and this should also be the breadth of the white space between the outer 
and inner lines. 

No attempt should be made to make even a line so thick as this at one stroke of the pen, 
nor yet to cover the whole with ink at once, as is sometimes done by means of a brush, or 
worse yet, by " filling in " the space ruled by two finer hnes by the use of either the drawing- 
pen or a writing-pen. Such a river of ink not only requires a long time to dry, but is apt to 
cause the paper to "buckle," after which it is very unlikely to become smooth upon drying. 

The proper course is this : Rule parallel lines of moderate thickness a short distance 
apart ; when these are thoroughly dry, and not before, rule others between them, until the 
whole breadth is covered. Thus a smooth, black, and even line of any width may be 
obtained. 

21. Converging Lines. — When, as in Fig. 5, a number of lines are to be drawn radiating 
from a single point, it is clear that one will meet and 
merge into another at some little distance from the 
actual point of convergence. In such cases wait 
until each line is dry before drawing the next one, 
in order to avoid the accumulation of a mass of ink 
at the junction. And this will be avoided with the 
greater certainty by drawing the lines away from 
and not totvard the point from which they radiate. 

22. Fig. 6 is a sheet of exercises similar to those 
of Fig. 2, and no additional instructions are required, 
except in relation to the inner corners of the hollow 
rectangle, which are rounded, or, as it is technically 
called, " filleted." In pencilling, the sides of the inner 
rectangle are drawn first, and the circular arcs next 

drawn tangent to them with the bow-pencil. The centre c of such an arc, Fig. 7, A, can 
after a little practice be readily and accurately located by trial, after which, if it be deemed 
necessary, the points of contact, m, n, may be marked by aid of the T-square and triangle. 
If the radius of the circle is large, it will be advantageous to draw a short line at an angle of 
45° through the corner x oi the pencilled rectangle, upon which the centre must lie, as in Fig. 
J, B\ or to set off from x, as shown at E, the points ;;«, n, with the given radius, and around 
these points to describe short arcs with the same radius, intersecting in the required centre. 
But in drawings of machinery so many of these small fillets are met with, that such geo- 
metrical operations consume too much time. 




Fir,. 5. 



10 



MECHANICAL DRAWING. 










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ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 



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23. Ink in the circular arcs first and the right hnes last, because it is much easier to 
join the right line smoothly to an arc already drawn, than to join an arc smoothly to even 
one right line, let alone two — as may be readily 
proved by trial. In the case of a circular arc 
the reason is clear, for the least error in either 
centre or radius will produce a sensible effect 
at one or both the junctions. It may be said 
that such an error would impair the accuracy 
of the work even if the arc were inked in first ; 
this is true, but if the error be, as it often is, 
practically unimportant, there is no need to 
make it conspicuous, as it will be if the junction, 
as at Fig. 7, D, is careless or imperfect. 

The difference in the difificulty of making a 
continuous line is not so great if the curve be 
one for which a " sweep," or curved ruler, can be used ; but a systematic procedure is likely 
to save time, and straight-line work is preferably left to the last. 

24. Sectioning of Hollow Figures. — In Fig. 8 are shown transverse sections of a hollow 
rectangular beam and a hollow cylinder. In sectioning these at the angle shown, it is natural 
and right to begin at C. After passing the point E many think it necessary to section the 



D 



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Fig. 7. 





Fig. 8. 



opposite sides at the same time, drawing, for instance, the line ab, then the line cd, without 
moving the triangle. This involves a serious loss of time, if the surfaces are of any consid- 
erable magnitude. 

A better method is to section the upper side first, going on from cd to the right, stopping 
when a line mn is drawn, such that a half-dozen or so more would carry us to the point F. 
Then go back to ab, and section the other side — it will then be seen that when the edge of 
the triangle again coincides with mn, the error in the position of op cannot exceed half a 
space ; and by dividing this error up among the spaces yet remaining before reaching F, the 
individual variations will be imperceptible. The saving of time in the sectioning of a large 
cylinder by adopting this method instead of the other amounts to fully twenty-five per cent, 
after a moderate degree of facility in the manipulation has been attained. 



12 



MECHANICAL DRAWING. 



TANGENCY OF CIRCLES AND LINES. 
25. For the purpose of acquiring skill in the handling of the compasses, and habits of 
care in the execution of work at every step as well as in the final processes, no practice is 
better than that afforded by exercises in which it is required to draw a number of circles 
within a given figure, tangent to it and to each other ; some of which may now be appropri- 
ately taken up, in order to relieve the monotony of continuous straight-line work. 




Two such exercises are given in Figs. 9 and 10, the external figures being a circle and 
an equilateral triangle, within each of which three tangent circles are to be inscribed. 

26. To construct Fig. 9. 

1. Draw the vertical centre line and describe the larger circle about O. 

2. Without laying down the compasses, set the needle-point at D, and mark the points 
C, E, the radius being unchanged. 

3. Similarly, set the needle-point at A, and mark the points B, F. 

4. Still with the same radius, describe about B the semicircle OCG. 

5. Draw the diameters FC, EB, and produce the latter to G. 

6. Draw the tangent at A, and produce it also to G. 

7. Bisect the angle AGO by the right line GH, cutting AD at L. 

8. Describe about O a circle through L, cutting CF diwd EB at M, N. 

9. About L, M, N, with radius LA, describe the three required circles. 

27. To construct Fig. ID. 

1. Draw the vertical centre-line of indefinite length. 

2. Lay the scale against the T-square, set off DB = DA, and draw AB. 

3. Set the zero of the scale at A, and mark the point C on the vertical centre-line, mak.- 
ing AC = AB. 

4. See if CB is equal to AB ; and if correct, mark its middle point E. 

5. Mark with the scale the middle point F o{ AC. 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 13 

6. Draw AE, BF, cutting CD in O, the centre of the triangle. 

7. Bisect the angle AEC by a line cutting CD in G. 

8. Describe a semicircle on diameter AB : it should pass through G. 

9. Describe about O a circle through G, cutting AE, BF, in N and M. 

10. Draw GP perpendicular to BC ; it is the radius of the required circles. 

28. The above instructions have been made thus explicit in order to emphasize one or 
two points already mentioned ; more particularly, the doing as much as possible at once with 
one adjustment of the compasses, as in Fig. 9, and the use of the scale when practicable 
instead of the compasses, as in Fig. 10. It is very common to see the triangle laid out by 
striking arcs about two of its angles as centres: in doing this, holes are pricked in the paper, 
into which the ink will run, so that the sharp finish desirable is dstroyed. Use the scale. 

Another object is, to call attention to the advisability of cJiecking the accuracy of any 
step, when the greatest precision is required. Thus in Fig. 9 the point G is the intersection 
of the tangent at A with the prolongation of the diameter EB\ and the precision with which 
it is located is confirmed if it falls, as it should, upon the circumference OCG : evidently, an 
error here would be vital, and affect injuriously the final result. Another instance is found 
in Fig. 10, where the intersection of the two straight lines EG, CD should fall on the semi- 
circumference y^G^^. The application of such checks is particularly necessarj' in laying out 
diagrams of this description, for though geometrically simple, they are practically very diffi- 
cult to draw with perfect precision. 

29. Bisection of an Angle. — In Fig. 9 the arc NP is drawn, merely to indicate that the 
arcs intersecting at H,\\-\ bisecting the angle y4G"C, are described about centres P and A^, 
equidistant from the vertex G\ and thus in this illustrative diagram to make a record of the 
processes. Practically, the arc need not be drawn at all, it being sufificient to set off by the 
scale the two points Pand N, and the farther from G the better ; for the distance between 
H and G being thus increased, any error in the direction of the line determined by those 
points will diminish as the line approaches the vertex. Theoretically, the radius of the arcs 
intersecting at H is arbitrary ; but if they cut each other acutely, it is obviously difficult to 
locate the precise point of intersection. 

Not only in the bisection of angles, then, but in various other operations, the draughts- 
man should always keep in mind the two following important principles, viz.: 

30. I. Points for locating lines should be as far apart as possible. 

2. Lines for locating points should cut each other normally, as nearly as 
possible. 

31. The necessity of exercising great care at everj' step in constructing these and similar 
diagrams will be appreciated if we consider for a moment what tangency really means. 
Supposing the instrument lead to be sharpened, as it should be, to a keen edge, so as to 
draw a fine clean line, two tangent circles ought to appear as at A, Fig. 11 ; merging at the 
point of contact into one line without increased thickness. A very slight error in the radius 
or the position of the centre of either of the circles will lead to conspicuous faults, shown at 
B and C. That there should be no increase of thickness at ^, is clear from the consideration 
that the pencil (or ink) marks represent mathematical lines, which have no thickness ; so that 
two, or twenty, are no thicker than one. 



14 



MECHANICAL DRAWING. 




Fig. II. 

32. In inking in, however, a much more effective appearance may be given to such 
exercises by using lines of different thickness ; the centre-Hnes and construction-lines being 
very fine, while the given figure and the required tangent-circles are made much thicker. 
This is entirely distinct from the use of shadow-lines : all the heavy lines here referred to are 
made of uniform thickness, and perfectly smooth and even. In this way the data and the 
results, being made bold and prominent, strike the eye at once, thus better exhibiting the 
nature of the problem ; while the processes by which the results are reached, though clearly 
indicated, take their proper secondary place. 

Use of Broad Lines. — It is sometimes necessary to represent a line by a broad band, as 
for instance in making lecture-room diagrams, which must be distinctly visible to the whole 
audience. When seen from a sufficient distance, such a band will appear like a fine line, the 
position of which is mentally referred, not to one or the other edge of the actual stripe, but 
to its centre. Such diagrams should therefore be drawn at first with at least moderately fine 
lines, on each side of zvhich, at equal distances, parallel lines may be drawn, the space between 
them being subsequently filled in. 

33. It follows from this, that no matter how heavy the lines of the triangle and the 
circles in Figs. 9 and 10, they will be no thicker at the points of tangency than anywhere 
else. When viewed from a short distance, the lines no doubt seem to coincide for a sensible 
extent, instead of at a single point: but so would finer lines if put under a microscope ; and 
if the eye be sufficiently removed, these heavier ones will appear truly tangent, as in fact 
they are. Beginners are apt to think that two tangent arcs should, when drawn with broad 
lines, appear as in Fig. 12, A, because, as they say, the lines just touch each other. The 
fallacy will be clear if this method is applied to a reverse curve composed of two tangent 

A :b CD 





Fig. 12. 



arcs, as at B ; the effect is shown at C, one glance at which is enough, it being perfectly 
obvious that the result desired is that shown at Z>, which is produced as above explained. 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 



15 



34. Dotted Lines. — Some of the lines in Figs. 9 and 10 are Avhat are technically called 
" dotted ;" which strictly means broken, that is, formed of alternate short dashes and spaces, 
for true dots cannot be made with a drawing-pen, nor would they equally well convey the 
idea of direction, if they could be. 

In working drawings, dotted lines are used to indicate the outlines of parts concealed 
by intervening objects; but as to lines used merely in construction, there is no absolute rule 
for determining which ones should be dotted and which should not. But in all cases the 
dotting should be regular and even, the spaces just long enough to be distinctly seen, and 
equal in length to the dashes. If the dashes are long, no matter how regular, the effect is 
that of apparent haste ; and no one minor thing adds so much to the finish of a drawing as 
dotted lining which is fine, firm, clear, and well spaced. 

35. Figs. 13 and 14 are exercises precisely similar in character to those given in Figs. 9 
and 10. In Fig. 13 the centres and points of tangency divide the diameter into six equal 
parts : they may therefore be at once marked at one setting by means of the scale, by select- 
ing a suitable value for LM, and making LR six times as long. 




Fig. 13. 



In Fig. 14 construct the triangle by means of the scale, as in Fig. 10 ; bisect the angle 
ABF by the line BH, cutting the vertical CD in L, the centre of one of the required circles ; 
the other centres are determined by drawing the equilateral triangle MLNO. 

36. Judicious Arrangement of Diagrams and Drawings. — It is understood that, as 
intimated in (6), the border-line is to be first drawn, and the exercises arranged within the 
space thus enclosed. In order that the sheet when completed may present a neat and well- 
balanced appearance, due attention must be paid to the relative positions as well as the 
actual dimensions of the figures which compose it. The two pairs of diagrams mentioned in 
the preceding section suflficiently illustrate the fact that perfect symmetry is not always 
attainable ; — which is here emphasized, because experience has shown that beginners are very 
apt to think that when, as in these cases, two figures are to be drawn, the first thing to do 
is to draw a vertical line through the centre of the paper. Nothing could be more erroneous, 



i6 



MECHANICAL DRAWING. 



as the reader will at once perceive by imagining a border to be drawn, say, around Figs. 9 and 
10. Were the space within the border thus divided into two compartments or panels, the 
diagrams would have to be made so small as to be out of all proportion to so large a sheet. 
Should this dividing line be inked in, the effect would be more tolerable ; but if not, the two 
figures would, by reason of the great distance between them, appear as if they repelled each 
other, and were crowded out toward the border-line. 

The fact is, that one diagram is in a sense a border to the other; and even if, as in some 
of the following examples, they are perfectly symmetrical, it should be kept in mind that the 
space between the two need not be any greater than that between either of them and the 
adjacent end line of the border. 

37. Fig. 15 is an open-work eight-pointed star enclosed within two interlacing hollow 




Fig. 15 



squares. The latter are first drawn, and the breadth x evidently determines the positions of 
the points A, B, C, etc., of the star. The radiating arms Z, M, etc., might be laid out by 
first drawing a centre-line for each, and setting off the half-breadth on each side with the 

spacing dividers; but a better method is illustrated in Fig. 17. Draw 
in pencil a small circle at the centre of the star, of a diameter equal 
to the breadth of the arms, whose outlines are drawn tangent to this 
circle by means of the T-square and the triangle. The eye can judge 
with great nicety as to the fact of tangency; and this method will be 
found of great advantage in many similar cases. . 

38. Fig. 16 is another example of interlaced work, the furnia- 
mental figure being a five-pointed star. To locate the points of this 
star means to inscribe the pentagon in the circle, which is done as follows: 

1. Bisect the horizontal radius OA at C. 

2. About C describe an arc through D, the extremity of the vertical radius, cutting the 
horizontal diameter in F. Then DF — side of pentagon. 




Fig. 17. 



ELEMENTARY EXERCISES : STRAIGHT LINES AND CIRCLES. 17 




i8 



MECHANICAL DRAWING. 



The centres of the circular arcs forming the inner interlacing figure are the intersections 
J, L, etc., of the outer lines of the star. The radius of all the inside arcs is equal to half^Z; 
these arcs are therefore tangent to each other at R, P, etc.; and great care must be taken in 
manipulating the shadow-lines, in order to produce the sharp knife-edge finish essential to a 
proper efTect. 

39. In the drawing of interlaced work it is clearly impossible to leave out in 
pencilling all lines which are not to be inked in, because it cannot be known just which 
ones they are. Since then all the lines must be pencilled in at first, crossing each other as 
they may, this should be done very lightly ; after which it is advisable, as a precaution 
against mistakes with the pen, to erase carefully with the sharp corner of a small piece of 
rubber those portions which are not to be drawn in ink. Work of this description gives 
good training in the use of the pen, since, in order to avoid a ragged appearance of the draw- 
ing, the lines must meet sharply without overrunning ; its perfect execution requires both 
care and skill. 

40. Fig. 1 8 represents a completed sheet, in which the three upper figures are simple 
exercises in interlaced work ; the locations of the centres of the circular arcs being indicated, 
no explanation is necessary. The lower figure is a panel ornamented with fretwork in the 
Greek style, the panel being sunken and the fretwork raised ; the effect being made 
pronounced by the use of quite heavy shadow-lines. This pattern, though very simple, is 
yet of such a nature that the attempt, by using the scale for each member and part of a 
member, to end every line in pencil just where it is to end in ink, would be clearly absurd. 

Rule the horizontal lines very lightly from end to end, as indicated by fine lines in Fig. 19 ; 







































- 






















. .. . 






























































































































































































— , — ,_, — , — . — .-_ 






_ 





Fig. 19. 

lay the scale on the lower line, and with a sharp pencil set oH the longitudinal spaces; then 
by means of the triangle the vertical lines can at once be drawn of exactly the right length. 
After this is done, it will be safer to pencil in a little more strongly, by means of the T-square, 
those portions of the horizontal lines which are to be drawn in ink; without this precaution 
mistakes with the pen are likely to occur. 

41. Fig. 20 is an exercise which, while apparently simple, is yet a very difficult one to 
execute perfectly. The diameter of the outer circle being divided into any number of equal 
parts, begin at one end, and describe a semicircle upon each part, on the same side of the 
diameter. Then, beginning at the other end, do the like in reverse order on the opposite side 
of the diameter. Thus at each extremity of the diameter a number of arcs are tangent to 
each other; in order to prevent blotting, begin with the outer one, and wait until it is dry 
before drawing the next. This order is also best for another reason — the lines, having some 
thickness, actually meet each other before reaching the point of tangency ; hence the pen 
need not always start from that point, but may first touch the paper where the lines coalesce. 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 19 

Moreover, the curvilinear angles become less and less toward the outer part, so that the 
daylight between the arcs should extend farther in toward the point of tangency, as the radii 
diminish ; this requires close attention, and can be better controlled by following the above 
order in drawing the semicircles. 

Instead of assuming at random a radius for the outer circle and using the dividers for 
subdividing it, thus marring the paper, it is advised to set off an assumed unit the requisite 




Fig. 20. Fig. 21. 

number of times by means of the scale and pencil, thus locating at once both the centres and 
points of contact of the various semicircles. 

42. Fig. 21 is an exercise of analogous character, in which the subdivisions on the 
horizontal diameter may also be most advantageously set off by using a scale, as above 
explained, the diameter itself of the outer circle C being also thereby determined. 

This being done, it is clear that the centre of the circle D, which must be tangent to the 
three circles A, B, and C, must lie on the .vertical diameter of C. The radius of D, and the 
location of its centre, may best be found by trial and error: this will give a result practically 
as accurate as that of a geometrical construction, which may be made, but the process is long 
and tedious. The diameter of D must then be subdivided, by the scale if practicable, and if 
not by means of the dividers, in order to locate the centres of the remaining circles. 

43. In Fig. 22 the horizontal and vertical diameters may be drawn with the T-square, 
and the incHned ones with the 45° triangle; but the work should afterward be carefully tested 
with the dividers, to make sure that the outer circle is accurately divided. 

Produce IK to cut at B the horizontal tangent through A, making AB equal to AO. 
Then a line bisecting the angle ABO will cut AO in C, the centre of one of the larger circles 
required. Draw through (7 a horizontal line cutting OK in E : then CE = CO, and E is the 
centre of one of the smaller circles, each of which is tangent to the two adjacent larger ones 
as well as to the exterior circle. 

44. The equilateral Gothic arch. Fig. 23, is drawn by describing about A and B, the 
extremities of the base, two circular arcs with radius AB, which intersect at C on the vertical 



20 



MECHANICAL DRAWING. 



centre-line CD. The problem is to inscribe three tangent circles in this arc ; the two lower 
ones will evidently be equal, and the centres may be found as follows : Set off on the 
prolongation of BA a distance measuring 3 on any scale of equal parts ; at E set up a 



B A 










// ' \^*'***» 

1. / y\ 


-^-^7 \ ^ 


M. 




^s/ ^^^ J 






Fig. 22. 



Fig. 23. 



vertical EF, measuring 2 on the same scale, and draw ED cutting the arc AC at G. Draw 
GB, and also a line perpendicular to it at G (which is therefore tangent to the arc AC^, and 
cutting DE at H. On ED set off HI = HG, and at /erect a vertical line cutting GB at L, 
the centre of one of the required circles. As a test, bisect the angle ADC by the line DJ, 
which should also pass through L. Draw through L a horizontal line cutting CD at M, and 
on it set off MN =^ ML: then yV is the centre of the second lower circle. The radius of the 
third circle will be less than that of the others : its centre will obviously lie upon CD, and 
both centre and radius are practically best found by trial and error. 

45. Fig. 24 is a panel ornamented with interlaced work, in which the pattern is composed 
wholly of circular arcs. The radii are uniform throughout, and it is apparent that all the 




Fig. 24. 

centres may be readily set out by using the scale alone as before described. This exercise is 
an excellent one for acquiring control of the compasses, since in order to produce a good 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 21 

effect the shadow-lines must be of uniform strength as well as nicely graduated, and as in all 
work of this description the lines must meet without overrunning. 

46. Figs. 25 and 26 are further exercises in " confined tangencies." In Fig. 25, having 
drawn the circle O and the trisecting centre-lines, we may draw the circles A, B, and C, with 




Fig. 25. Fig. 26. 

any assumed radius as aP. Then the circle D must touch the circle O at P, and also be 
tangent to B and C; its centre, which must lie on aP, as well as its radius, may best be found 
by trial. Fig. 26 is very similar in its nature ; but in this case the radius of A is so large that 





Fig. 27. Fig. 28. 



its centre a lies within both B and C. Another circle may therefore be drawn about a, 
internally tangent to B and (T, whose radius may be found by drawing ab and producing it 



22 



MECHANICAL DRA WING. 



to cut the circumference of A. Finally, by trial we find on aP di centre g, and a radius such 
that the circle G is tangent to O at the point P, and also touches the circles E and F, 
which are equal to D, and concentric with B and C. 

47. Figs. 27 and 28 are merely fanciful designs of interlaced work, in which the centre- 
lines, which are necessary of course in laying out the work, are not shown. In regard to the 
former, it is sufificient to say that the centres of the three circles are at the vertices of the 
inner equilateral triangle, and the compasses must therefore be handled very lightly in order 
to avoid making unsightly holes in the paper and thus marring the finish. In Fig. 28 the 
vertices of the interlacing curvilinear triangles must of course lie at six equidistant points 
upon the circumference of a circle, and the centres of the arcs which form their sides lie 
upon the prolongations of the diameters passing through those points. The radii of these 
arcs is arbitrary: indeed, much in relation to the proportions and details of these exercises is 
purposely left to individual discretion, because training in judgment and forethought is quite 
as essential to success as that in manual dexterity. In both these exercises the broad black 
lines are to be drawn as explained in (20); and in Fig. 28 special care in inking in is neces- 
sary as to the six outer points, particularly in manipulating the shadow-lines upon these 
acutely intersecting arcs, which should not appear blunted, but terminate in knife-edges, 
sharp, clean, and well defined. 

48. In Fig. 29, having described the outer and the inner circles, whose diameters are 
arbitrary, about the common centre O, divide the circumference of the latter into any number 
of equal parts at A, B, C, etc., and draw tangents to it at the points of subdivision. 



E 


X 




•^ 


( 




T? \ 


-^ 


^^/\ /^\- 




N^ 


/ 


yh\/-i^ 


\ "'^ 


^/^ > 


/ 


' ^1 


^r---^ 


\ 


\ 


/ 


\ /' 




-^ 


\ 

\ 


/ 


/^^_, 


, 




i^? 


\ 


^ 


TS. \/ 




>i 


A 


^^ 




V/ 


7 


\ 7/ 


/ 


^ \ 


>< 




y^ 


k / 


% 




\ 


-\. ' 


\ 1 


« 




\ N 


\ 






\ A 


> 


V. ^^ 




V .-' // 




X 




W."' 1/ 




X. 


. 


i"" 




-r \ 


. y 





"-^ 




c 









Str-"'/ 


^/^ 






>B K 


V 


B 



Fig. 29. 



Fig. 30. 



The tangent AK, produced to the right, intersects at D the tangent BI\ bisect the angle 
KDI by the right line DE, on which find by trial the centre, R, of a circle which shall be 
tangent to the outer circle P, and also to the right lines AK, BI. A circle described about O 
through R contains the centres of all the small circles similarly tangent to P, which of 
course are equidistant from each other. 

In Fig. 30, having drawn the outer circle /*, and the vertical centre-line AOB, describe the 



ELEMENTARY EXERCISES: STRAIGHT LINES AND CIRCLES. 



23 



circle D tangent to P, with any radius CA taken at pleasure ; bisect LB at E, which is the centre 
of circle F. Then find by trial the centre G, and the radius, of a circle //, tangent to P, D, 
and F\ also of a circle K, tangent to P, D, and //, and so on as far as may be desired. It is 
easy to show that the centres of all circles tangent to P and D lie upon an ellipse, of which 
O and C are the foci, and AE is the major axis ; b^t even the drawing of this ellipse would 
be of no practical advantage. 

49. Outlines composed of Circular Arcs. — In dr-awings of mechanical subjects it is 
frequently desirable to make up curved outlines of a series of circular arcs of different radii, 
'n order that the workman may readily copy them with the tools which are always at hand. 
This will in many cases give an approximation, sufificiently close for practical purposes, to a 
forni which it would require much more time and trouble to lay out with perfect accuracy. 

In Fig. 31, EFGH is a rectangle divided hy AC into two equal squares. In each square 
draw the diagonals, intersecting at B and D. About A as a. centre describe the arc EF; 
about B describe the arc EG, then about C and D describe the arcs GH, HE; thus forming 




Fig. 31. 



a closed figure, approximating in form to an ellipse. Using the same centres, with radii 
increased by a certain amount, this figure is surrounded by a narrow band of uniform width. 
Repeating the same operations with still greater radii, but keeping the same centres^ the result 
is a representation of a picture frame with two raised mouldings. 

It is clearly necessary that these various arcs should be truly tangent to each other, and 
the junctions imperceptible, as though each outline had been drawn by one continuous 
motion of the pen. 

Relief is given by the use of shadow-lines : in regard to which it may be said that prac- 
tically the portion LEK may be made of uniform thickness, the tapering of the line being 



24 



MECHANICAL DRAWING. 



confined to the half-quadrants LF, KH\ and the same holds true in relation to the other 
shadow-lines upon the mouldings. 

50. Fig. 32 shows an approximation to an oval outline by means of circular arcs. 
Describe about C a circle cutting the horizontal diameter at A and B and the vertical diam- 
eter at D\ draw AD, BD, and produce them. About ^ as a centre, with radius AB, describe 
the arc BF\ about B with the same radius describe the arc AE, and complete the inner oval 
by drawing the arc EF about centre D. The outer oval is drawn by means of arcs about the 
same ceni7'cs, but with radii increased by a definite amount assumed at pleasure. The space 
between the two ovals may then represent the outer wall of a sewer, and " sectioned " 
accordingly, the effect being heightened by the use of shadow-lines. As in the preceding 
example, the shadow-line may be made uniform from A to K, and tapered only from K to /, 
and from A to E : and similarly with regard to the exterior outline. 

TENTATIVE AND MECHANICAL PROCESSES LEGITIMATE. 

51. Before going farther, a little more may be said about the methods employed, to 
which allusion was made in (3). In some of the preceding explanations determinations by 
" trial and error " instead of by geometrical construction have been advised, which may to 
some appear at first glance inconsistent. 

But in drawing we have to do only with the representatioJts of magnitudes : thus a mathe- 
matical line, which has no actual thickness, must be represented by a mark of perceptible 
breadth. And the practical limit of accuracy is measured by the actual breadth of the 
finest visible line. Again, the ultimate test of the precision with which a geometrical 
operation has been executed lies in the answer to this question : Are the results which should 
follow actually obtained ? To illustrate : Let it be required to draw a circle tangent to three 
given circles. The centre and radius may be determined by a geometrical construction ; but 
if, when this has been made, the required circle sensibly fails to touch the given ones, it 
follows that an error has been committed. On the other hand, if by trial a circle be drawn 
which does touch the others, that fact proves that the centre and radius are those which 
should have been found by the construction ; they may therefore be accepted as correct. 

52. No positive rule can be laid down for determining which method should be adopted ; 
but if the geometrical operations are complicated, or the conditions such as to make any of 
the steps dubious, owing to acute intersections or like contingencies, it is easy to see that 
their results may be less reliable than those of the tentative processes above suggested. 

Tiie use of the latter is evidently legitimate in laying out working drawings and the like, 
.since in general the result only is to be represented. But even in making a diagram for the 
express purpose of illustrating a process, it is perfectly fair to make sure of the result by such 
means first, and to introduce the proper construction-lines afterward. Again, the draughts- 
man has often to perform operations for which there is no geometrical construction, as, for 
instance, in drawing a wheel with thirty one teeth : he must divide the circumference of his 
circle into the given number of equal parts, and this can be done only by trial. Even if the 
subdivision can be made geometrically, as in constructing the pentagon. Fig. 16, the final test 
is by trial, to see that the chord is correct ; and so, in general, the one method may be used 
as a ciieclc upon the other. 



ELEMENTARY EXERCISES : STRAIGHT LINES AND CIRCLES. 25 

53. Closely akin to these are certain mechanical methods constantly employed by every 
expert draughtsman. It need hardly be said that he who is possessed of two trustworthy 
triangles must also be possessed of an evil spirit if he does not use them for drawing lines 
parallel or perpendicular to each other, instead of resorting to a geometrical construction. 
This is no more mechanical than laying off an angle by means of a protractor : in either case 
the precision of the work depends upon the perfection of the instruments and the skill with 
which they are used, and indeed the same may be said of drawing a straight line through two 
given points by means of a ruler. In doing this the edge of the ruler is set, not so as to coin- 
cide with either point, but at a small distance from each, the eye being able to judge with 
perfect certainty of the equality of these distances, or as it is technically expressed, of the 
" daylight," between the ruler and the points. And this judgment is equally reHable if one 
or both the points be replaced by a curve. In this way it will be seen that if it is required 
to draw a tangent to a given curve through a given point or in a given direction, or a tangent 
to two given curves, the operations may be performed with the ruler alone with as much 
precision as though the points of contact were determined by geometrical processes. 

54. It may be accepted, then, that the eye can judge of \.\\^ fact of tangency between a 
curve and a right line when the direction of the tangent is either assigned or fixed by some 
other condition, with a degree of accuracy equal to that attainable in any graphic operation 
whatever. And no argument is needed to show that it would be absurd not to take advan- 
tage, in drawing mechanically, of a fact which in numberless cases enables the draughtsman 
to save both time and trouble, without in the slightest degree affecting the reliability of the 
results. But it is to be understood that the point of tangency cannot be located by the eye 
with any precision, and must in all cases be determined by some process of construction. 



26 



MECHANICAL DRAWING. 



CHAPTER II. 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 

55. It is often necessary, particularly in drawings of machinery, to construct with 
accuracy various matliematical curves ; and some preliminary practice in this kind of work 
being desirable, the following chapter is devoted to explanation of some of the more important 

of these curves, and of various graphic processes 
relating to them. In these and other operations 
there is frequent occasion to rectify circular arcs, 
and to lay off, either upon circles of given radii, or 
so as to subtend given angles, arcs of given linear 
values. 

In such cases the following constructions, due 
to Prof. Rankine, are extremely convenient ; the 
results are obtained much more expeditiously than 
by computation, with a degree of accuracy amply 
sufficient for all ordinary purposes. 

56. I. To lay off on a right line a distance approximately equal in length to a given cireular 




Fig. 33. 



arc. 



Draw the chord BA, and 

-l-e 





Let AB, Fig. 33, be the given arc, and AH its tangent at A 
produce it ; bisect AB in D, and set off AE equal to AD. 
About E as a centre, with EB as radius, describe an arc cutting 
AH \n F: then AF = arc AB, very nearly. 

II. To lay off on a given circle an arc approximately equal in 
length to a given right line. 

In Fig. 34, let AB, the given line, be tangent at A to the 

given circle. Set off AD = \AB, and 
about Z* as a centre, with radius Z'i? = 
\AB, describe an arc cutting the given 
circle in F. Then arc AF = AB, very nearly. 

III. To find the radius of an arc which shall subtend a given 
angle, and be approximately equal in length to a given right line. 

'Let AB, Fig. 35, be the given right line. Draw AG perpen- 
dicular to AB; and also AH, making the angle BAH equal to half 
the given angle. Set off AD — lAB, and about centre D, with 
radius DB = iAB, describe an arc cutting AH in F. Bisect AF by 
the perpendicular £(r, which will cut AG in (T, the centre of the 
required arc AF. 
57. Amount of the Error in the above Processes.— According to Prof. Rankine, the 
straight line is a little less than the arc in the application of either of these rules. He o-ives 



Fig. 35. 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 27 

the magnitude of the error as about -jig- part of an arc of 60° ; but it varies as the fourth 
power of the subtended angle, and may be reduced to a very minute amount by subdivision. 
Thus, for an arc of 30°, the error will be ■^\-^ X tV = tttto ! ^"^ fo'' ^"^ °f 20°, only -^\^ X 
gL = y^Iq 0^. If, then, the arc, given or required, subtends an angle of over 60°, subdivision 
should be resorted to in practically applying either process. 

The first two rules can also be applied to other curves than circles, provided that the 
change of curvature in the part to be dealt with be small and gradual. 

THE ELLIPSE. 

58. First Method. — About the centre C, Fig. 36, describe two circles whose diameters 
are respectively equal to the major and minor axes, AB and MN. Draw any radius at 
pleasure, cutting the inner circle in D and the outer one in E. Draw through D a parallel to 
AB, and through E a parallel to MN : the intersection O of these lines will be a point upon 
the ellipse. This is the most accurate of all methods for constructing this curve by points, all 
the intersections, D, E, O, being right angles. 

To draw the tangent at a given point tipon the curve. — Let P be the point : then by 
reversing the above process we find that P would have been determined by the radius CGH. 
At G and H draw tangents to the two circles, cutting the corresponding axes produced, in L 
and R, respectively: then LR is the required tangent at P. 

Conversely, to find tJic point of tangency. — Suppose it required to draw a tangent through 
a given point Q, or parallel to a given line XY. The tangent is drawn mechanically as 
explained in (53) : it cuts the axes at R and L. From R draw a tangent to the outer circle, 
from L a tangent to the inner circle ; these should be parallel to each other, and the points of 
contact are found by drawing CG^// perpendicular to both : then through G draw a parallel to 
AB, and through H 2, parallel to MN, which two lines will intersect in the required point of 
tangency P on the ellipse. 

59. Cautionary Remarks. — Particular attention is called to these points, viz. : 

1. The radii, CE, etc., need not be equidistant, and no attempt to subdivide the circle 
should be made. 

2. In construction, no radius should be drawn at all ; for instance, set the triangle as if to 
draw CK, but mark only the points /, K: then very short lines only, intersecting at T, need 
be drawn. See (4) page 2. 

3. The points determined should be nearer together where the curvature of the ellipse is 
most rapid ; but the actual number is arbitrary. This applies to the construction of any 
curve by locating points. 

4. For illustrating the process, two radii are sufficient : these should be made more 
prominent, as in the figure, than the perpendiculars and horizontals leading to the inter- 
sections upon the ellipse, which may be indicated by short full lines. 

60. Second Method. — Let C, Fig. 37, be the centre, AB the major axis, DE the minor 
axis. About D with radius AC describe an arc cutting AB in the foci F, F' . About i^ as a 
centre describe an indefinite arc with any radius FH, greater than AF and less than BE. 
About F' describe another arc, with a radius F' O = AB — FH : this arc will cut the one first 
drawn in 0, O' , two points of the required curve. 



28 



MECHANICAL DRA WING. 




EXERCISES IN THE DRAWING OF NON-CIRCUIAR CURVES. 29 

This method is not ehgible for the ordinary purposes of the draughtsman, because it 
involves much more labor than the preceding one, and the intersections of the arcs are in 
many cases too acute to be reliable. But it is of interest as depending upon the property of 
the ellipse that the sum of the focal distances is the same for every point upon the curve, 
and equal to the major axis; thus, 

PF-\-PF' = OF^ OF' = AF^AF' = BF-^BF' ^ AB. 

61. Third Method. — Upon this property, also, depends the operation of drawing the 
"gardener's ellipse" by the aid of a string and two pins. Let two fine pins be fixed in the 
drawing-board at F, F', Fig. 37; around these pass a loop of waxed sewing-silk, of which the 
total length is A B -\- FF ' : if this loop be kept constantly taut by a pencil P, the latter in 
moving will trace the ellipse. 

To drazv a tangent at any point, as O: Produce FO and F'O, and bisect the exterior 
angles FOI, F'OK by the line TT. To draw the normal at the same point, bisect the 
angles lOK, FOF' by the line NN. Obviously the axes cut the curve normally at their 
extremities A, B, D, E. 

62. Fourth Method. — Fig. 38 illustrates the principle of a common elliptographic 





1 
/■ 






A[ 


A ^ 


^ 


^/ V 


W 


^ 


C 








E 





Fig. 38. 
trammel. The three points P, M, N are in one right line, the distance PM being equal to 
CD, and PiV equal to CA. Keeping J/ always upon the line of the major axis, and jV upon 
the line of the minor axis, the point P will at all times lie upon the ellipse. 

This method is extremely convenient when the greatest precision is not required ; the 
three points being selected upon the graduated edge of a scale, or marked upon the edge of 
a smoothly cut strip of paper. 

To draw the normal at P. At M draw a perpendicular to AB, at iV a perpendicular to 
DE : these perpendiculars intersect at O, and OP is the normal required ; and this is inde- 
pendent of the position of the point P. 

63. Fifth Method. — To inscribe an ellipse in any given rectangle, Fig. 39. Join the 
middle points of the opposite sides by the right lines AB, EF: these will be the axes, and 
intersect in the centre C. 



30 



MECHANICAL DRAWING. 



Divide the semi-minor axis (7/% and the half side GF oi the rectangle, into the same 
number of equal (or proportional) parts. Through the points of subdivision on CF dra.w right 




Fig. 39. 

lines from B, and produce them to intersect the lines drawn from A to the correspondino- 
points on GF. These intersections will lie upon the ellipse. 

Or, divide the semi-major axis AC and the half side AH in like proportion, and proceed 
in a similar manner, the two series of intersecting lines being drawn from E and F, the 
extremities of the minor axis. 

64. The same process is applicable when it is required to inscribe an ellipse in any o-iven 
parallelogram, as shown in Fig. 40 ; but in this case AB, EF are not the axes. 




Fig. 40. 

They are, however, conjugate diameters, for each is parallel to the tangents at the 
extremities of the other ; and since the parallelogram can always be constructed if AB and 
EF are given, we have thus a simple and ready method of constructing the ellipse upon 
any pair of conjugate diameters. 

In order to determine the direction of the axes, describe about the centre C a circle 
upon either of the conjugate diameters, as EF: the circumference cuts the ellipse in P, and 
the supplementary chords PE, PF are parallel to the axes. 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



31 



65. Sixth Method. — To construct the ellipse by means of ordinates of the circle. In 
Fig. 41, let it be required to draw the ellipse of which CP is the semi-major and CR the semi- 
minor axis. 

Describe a circle with radius AE ^ CR, and divide CP and OA into any number of pro- 
portional parts. At each point of subdivision erect a perpendicular to CP, equal to the 




Fig. 41. 

ordinate of the circle at the corresponding point of OA : the curve PR thus determined is the 
required ellipse. 

Or, AE may be made equal to the semi-major axis, as BE: then if OA and the semi-minor 
axis ^^Vbe similarly subdivided, the ordinates of the circle will be equal to the corresponding 
ordinates of the ellipse N^E. 

66. In a given ellipse, to find the conjugate to a given diameter. 

In Fig. 42, let PO be the given diameter. Draw any chord EF parallel to PO, and bisect 
it : then the required conjugate diameter Jl/N passes 
through the point of bisection, and TT, the tangent at P, is 
parallel to MN. 

OtJierwisc : Draw the chord EE parallel to PO, and 
also the diameter EG : then AIiV and 7T are parallel to the 
supplementary chord EG. 

67. It is not, of course, intended or expected that an 
ellipse should be drawn by each of these various methods 
merely as an exercise : it is proper, however, to illustrate 
them here, since all these different properties of the curve will be found applicable at one 
time or another in subsequent operations. 

In making the constructions for practice only, it is recommended that two ellipses should 
be drawn on each sheet, the major axis of one being horizontal, that of the other vertical ; 
that these should be laid out by different methods ; and particularly that they should be made 
of quite different eccentricities: for while the law of the curve is always the same, the actual 
form is capable of infinite variation, and the course here advised will give greater practice in 
the use of the sweeps. 

Practical Suggestions in regard to the Drawing of Curves. 

68. A sufficient number of points having been determined, the curve should first be 
sketched in very lightly with the free hand ; if there be any serious error in the location of 




Fig. 42. 



32 MECHANICAL DRAWING. 

any points, it is likely to be thus detected. The contour should then be pencilled in more 
firmly by aid of the " sweeps" or curved rulers, which also are used in finally drawing it with 
the pen. The pencil line should be fine and clear; if the ink line be made heavy, half its 
thickness should be outside and half inside of the exact contour: see (32) and (33). 

Approximating Circular Arcs. — It is not only legitimate, but advisable, to make use 
of the compasses in inking in. By trial a radius may be found with which an arc can be 
drawn, having its centre upon the major axis, vv'hich will agree so closely with the ellipse for 
some distance on each side of the vertex, that the difference is absolutely imperceptible even 
when the line is as fine as possible. Thus in Fig. 38 the arc ee may be drawn about the 
centre d; and in like manner JiJi, is an arc described about the centre^, on the line of the 
minor axis. The same radii being used for arcs through the opposite vertices, perfect sym- 
metry is ensured, and there remain only four portions equal to eli, to be drawn with the 
sweeps. The points, ce, Jili, etc., should of course be previously set off, equidistant from the 
vertices, to secure equality of length in the opposite arcs. 

This process is also applicable to many other curves, as the parabola, hyperbola, sinusoid, 
etc., which are symmetrical about an axis. 

69. In the case of a very long and narrow ellipse, the centre corresponding to ^ in Fig. 38 
may become inaccessible ; and the part Iih of the curve must also be drawn by means of the 
sweeps. It is a very common error to suppose that in any case accuracy can be secured by 
adjusting a curved ruler to draw from E to // on one side, and then reversing it to draw from 
E to 1i on the other. Even if the tangent at the vertex be first drawn in pencil, the attempt 
to do this will almost certainly result in the formation of a "hump" or "broken back" at E. 
The proper course is indicated in Fig. 39. Draw the tangent at E: then about i; as a centre 
describe arcs cutting both the curve and the tangent on opposite sides of the vertex, as at 
a, a, b, b. The eye can judge accurately of the equality of the corresponding intercepts, and 
the sweep can thus be adjusted to draw at one stroke of the pen an arc of the curve extend- 
ing some distance each way from the vertex. 

70. In using the sweeps, certain precautions are requisite in order to make a smooth 
and " fair" line. Beginners are apt to set the sweep so as to draw a line through three or 
four points, and in readjusting it, to begin at the last point of the line just drawn, and set it 
so as to agree with the next three or four, and so on. The result is likely to be painfully 
conspicuous by reason of "humps" — the various sections of the curve thus traced not being 
truly tangent to each other. To avoid this : i. Never draiv quite as far as yon think you can 
at one setting of the sweep ; in other words, stop before the sweep deviates sensibly from the 
curve. 2. In readjusting, always set the sweep so as to coincide for a little distance with the 
portion of the curve already drawn. 

71. Copying by Transferring or Impression. — The ellipse is composed of four simi- 
lar and equal arcs, or quadrants. If one of these has been accurately laid out by any of the 
methods described, the labor of constructing the other three may be saved by the following 
expedient : Suppose, for instance, that the quadrant AE, Fig. 39, has been drawn with 
precision : upon a piece of thin transparent paper make a careful tracing of it with a sharp- 
edged, moderately hard pencil, and trace also the axes y^C, iT^T. Turning this paper over, 
and adjusting it by these centre-lines, go over the visible; contour again with a firm pressure : 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



33 



this will leave a faint trace of the quadrant EB, and will also make a mark on the reverse 
side of the thin paper ; which by like treatment will yield an impression of the quadrant BF ; 
and FA may be also similarly transferred. This simple proceeding can be applied in the case 
of many other curves composed of similar and symmetrical branches, as well as in the copy- 
ing of others which are to be reproduced in several different positions. Much labor in the 
way of construction can thus be saved ; but in the various tracings the sweeps must be used 
as conscientiously and as carefully as though the results were final. 

THE PARABOLA. 
72. The Parabola, Fig. 43, is a curve every point of which is equally distant from a 
given point F, called the focus, and a given right line DD, called the directrix. It is there- 
fore symmetrical about the axis OFC perpendicular to DD, and its vertex V lies at the 



\ 




!® // 


\li 


J- -^N^ 




V A \ 


•--_,\. 


A 



Q 

Fig. 43. 

middle point of FC. Draw any line parallel to DD, as RLP: then the points in which this 
line is cut by an arc described about F, with a radius equal to LC, will lie upon the curve. 

To draw a tangent at any point, as P : Draw FP, and also PA perpendicular to DD : then 
the required tangent bisects the angle FPA, and the normal bisects the angle FPB. 

Otlicrzvisc : Let R be the given point. Draw RL perpendicular to the axis, and on the 
axis set off 'VM = Z F: then MR is the required tangent. 

To drazv a tangent through a point witJioiit the cnrvc, and find the point of tangency : Let 
Q be the given point. Draw the tangent mechanically as explained in (53), and produce it to 
cut the axis in M ; set off on the axis VL = VM, and through L draw a parallel to DD\ this 
will cut the tangent in the required point R. The same process applies to the drawing of a 
tangent in a given direction. 

73. Second Method. — In Fig. 44, let V be the vertex, VO the axis, and P a point 
through which the curve is to pass. Draw PR perpendicular to VO, and make QR = PQ. 
On the axis set off VL = VQ\ draw PL and RL, divide them into the same number of equal 
parts, number the points of division in opposite directions, and join the points correspond- 
ingly numbered, as i, i, 2, 2, etc. : the lines thus drawn will be tangents to the required curve. 



34 



MECHANICAL DRAWING. 



To find the point of tangency on any one of these lines, for instance i, i. This line cuts 
the axis at N\ set off on the axis, VK— VN, and draw KB perpendicular to VO \ this will 
cut I, I, in the required point B. 




Fig. 44. 

74. Third Method.— In Fi<j. 45, let Fbe the vertex, VO the axis, and Pa point through 
which the curve is to pass. Draw PO, VA, perpendicular to the axis and equal to each other, 
and join AP. Divide VA and AP into any number of proportional parts ; through the points 

A; ^ ^ ^ 




Fig. 45- 

of division on VA draw parallels to the axis, and from those on yiPdraw lines to the vertex : 
the intersections of lines thus drawn through corresponding points on VA and AP will lie 
upon the parabola VP. 

To find the focus and direetrix : Draw a tangent to the curve, making an angle of 45° 
with VO (72) ; this cuts the axis at the point C of the directrix DD, which is perpendicular to 
VO. On the axis set off VF = VC : then F is the focus. 

75. Application. — A pleasing application of this curve to practical purposes is found in 
the pointed arch, Fig. 46: AC he'mg the span and BD the rise, the outer line of the arch is 
composed of the two parabolas AB, BC, of which the common axis is AC, and the vertices 
are A and C. These may perhaps be best constructed by the method of Fig. 45, as indicated 
by the rectangle AEBD, and the two lines intersecting in x. 

The inner curves ah, cb are not true parabolas, but parallels to AB, BC: that is, the 
normal distance between the outer and inner curves is everywhere the same. These interior 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



35 



curves are best drawn thus : Strike with the bow-pencil a series of arcs with a radius equal to 
the assumed breadth Aa, all the centres being on the parabola AB. Then by means of the 
sweeps draw the envctope of these arcs, that is, a line tangent to them all. 



E 




B 




-tj^^r^ 


^^^^^^^^\ 




1 

f 


r 




\ 


A 




a. 


D c 




C 



Fig 46. 

Note. — Points on this interior curve might be found by drawing normals to the parabola 
AB by methods previousl}' indicated, and setting off on each the assumed distance Aa. But 
the process just described is far less laborious, and if the number of arcs be equal to that of 
the points, it is quite as accurate, if not more so. 



THE HYPERBOLA. 



76. The Hyperbola is a plane curve generated by the motion of a point subject to the 
condition that the difference of its distances from two fixed points called the foci shall always 
be equal to a given line, whose length must be less than the distance between the foci. 

In Fig. 47 set off on the horizontal line CF = CF' , and let F, F' be the foci : also set ofi 




Fig. 47. 

CA = CB, and let AB he the given constant difference. Clearly, then, FB — BF' = 
AF' — AF = AB \ therefore A and B lie on the curve. With any radius /^C greater than 
FB, describe an arc about i^ as a centre: then about F', with a radius F'O = FO — AB, 



36 



MECHANICAL DRA WING. 



describe another arc, which will cut the one first drawn in two points O, O', of the hyperbola. 
Since with the same radii arcs may be described about the other foci, it follows that the 
curve has two equal and opposite branches; which are infinite, because /^(9 may be of any 
length. 

The point C is called the centre, and the line AB the major axis, to which the minor 
axis passing through C is perpendicular. 

To draw a tangent at any point of the airve, as P : Draw PF, PF', and bisect the angle 
between them ; the bisector is the required tangent. Otherwise t/tiis : Describe a circle upon 
AB as a diameter, let fall a perpendicular PG upon the major axis, and from its foot G draw 
a tangent to the circle. From K, the point of tangency, let fall a perpendicular KL upon the 
major axis; through its foot L draw LP, the tangent required. By reversing this process the 
point of contact may be found, when the tangent is drawn mechanically (72), through a given 
point not upon the curve, or parallel to a given line. 

To find the asymptotes : From either focus, as F', draw tangents to the circle described 
upon diameter AB\ find the points of contact R and 5: then CR and CS are the asymptotes. 

77. Second Method. — In Fig. 48, let C be the centre, AB the major axis, and O a 
point through which the hyperbola is to pass. Describe upon AB the semicircle ADB, and 




Fig. 48. 



draw DCE perpendicular to AB, also DT tangent to the semicircle. Draw C5 perpendicular 
to DE., and with radius OS describe about C an arc cutting Z^T" in Z; on SO set off 
SG = DL: then CG is an asymptote to the curve. Draw any line ^^7 parallel to AB, cutting 
CG in /, and produce it; set off on the tangent, DM ■= KI, then on the prolongation of KI 
set off KN = CM, and A^ will be a point on the hyperbola. 

To find the focns : Produce GC to cut the semi-circumference in R, at which point draw 
a perpendicular to GR; this will cut the line of the major axis in the focus F. 

To draw a tangent at any point on the curve, as P : Draw PV parallel to the asymptote 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



37 



HC, cutting the other asymptote GR in V. On GR set off VU = CV, and draw PU, which 
will be the required tangent. 

To find the point of contact when the tangeitt is given : Suppose PU to have been drawn, 
mechanically (72), tangent to the hyperbola. Produce the given tangent to cut one 
asymptote, GR, in U, and the other one, HC, in Q. Bisect QU m P, which will be the point 
of tangency sought. 

To draw the hyperbola through a given point, the asymptotes and the directions of the axes 
being known : This may be done by reversing the construction of Fig. 48. Thus, CG and 
CH being the given asymptotes, let it be required to draw the hyperbola through the given 
point N ; the major axis to be horizontal and the minor axis vertical. Draw NK per- 
pendicular to the minor axis CE, cutting the asymptote CG in /; through /draw a parallel 
to the minor axis, intersecting at M an arc described about C with radius equal to KN; 
through J/ draw a horizontal hne, cutting the line of the minor axis in D: then CD = serni- 
major axis, and the curve can be completed as before. 

78. Third Method. — Given, in Fig. 49, the major axis AB, and P a point in the 
required curve. Draw PO, BE perpendicular to AB and equal to each other, and join PE. 




Fig. 49. 



Divide PO and PE into the same number of equal (or proportional) parts, numbering the 
points of division from P on each line. From A draw lines to the points upon PO, and from 
i? lines to the points upon PE: the intersections of the lines thus drawn to corresponding 
points of division, as for instance Ai, Bi, will lie upon the hyperbola required. 



DRAWING OF ROLLED CURVES. 



79. Rolled Curves, Roulettes, or Epitrochoids, are curves described by points carried 
by one line which rolls upon another, the latter being fixed. The line which carries the 
tracing-point is called the generatrix or describing-line, and the fixed one is called the 
directrix or base-line ; either of these may be straight, or both may be curved. 

The nature of perfect rolling contact may be best seen by a study of that which is not 



38 



MECHANICAL DRAWING. 



perfect. The polygon in Fig. 50 rolls along the fixed right line with a hobbling motion. In 

the position shown, it is clear that if there is to 
be no sliding the point A must be at rest, and 
the motion indicated by the arrow must consist 
in a rotation of the whole figure about yi as a 
fixed centre, until B reaches M, and the side AB 
coincides with LM. The polygon will then turn 
about ^ as a centre, and so on ; its perimeter 
measuring itself off upon the right line. If the 
A- M. number of sides be increased, the hobbling will 

^'^- 5°- be diminished, and if the number becomes incon- 

ceivable, it will be imperceptible. The broken outline then becomes the dotted curve, to 
which the right line is tangent, and the change from one centre of rotation to another goes 
on continuously. But what is true up to the limit holds true at the lim.it, and the fact 
remains, that at any instant the point of contact is at rest, and every point in the figure is at 
that instant tjirning aboiit that point of contact as a fixed center. 

80, And upon this fact depends the readiest and most reliable known method of draw- 
ing the curves in question. In Fig. 51 A A is a curved ruler fixed to the drawing-board, 





Fig. 51. 

and BB is a free one rolling along it. Let a pencil be fixed to and carried by the latter, 
either in the contact edge as at D, or at any distance from it, as at E. Since P, the present 
point of contact, is the instantaneous centre of rotation, the point D is at the instant moving 
in the direction Z^/% perpendicular to the contact radius/?/'. Therefore /?/^ is tangent to 
the path of D, traced as the ruler BB rolls; but it is also tangent to the circular arc whose 
centre is P and radius PD, consequently the path of D is also tangent to that arc. 

Now, let the arcs Pc, Po, of BB, be equal to the arcs Pc', Po' , of A A : then cD will be 
contact radius when c reaches c', and oD when reaches 0'. If then we describe, with these 
radii, circular arcs about c' and 0', the curve traced by D will be tangent to those arcs; and 
that traced by E will be tangent to arcs about the same centres, with PE, cE, and oE as 
radii. 



EXERCISES IN THE DRAWING OE NON-CIRCULAR CURVES. 



THE CYCLOID, EPICYCLOID, AND HYPOCYCLOID. 
81. The Cycloid is traced by a point in the circumference of a circle which rolls upon 



its tangent. 



In Fig. 52 find the length Aa of a convenient fraction Aa of the circumference, 
(see 56); step this off the required number of times upon the tangent, making AE equal 
to the semi-circumference. Divide each into the same number of equal parts, draw 
chords from P to the points of division on the semicircle, with which as radii strike arcs 
about the corresponding points on AE : the cyloid is tangent to all these arcs. The more 
numerous the arcs, the more perfectly is the curve mapped out. 




Fig. 52. 

To find points on tJic curve: By the above process no points on the cycloid are 
located ; the curve touches all the arcs, but the points of contact are not known ; nor is it 
necessary, for the purpose of merely drawing the cycloid, that they should be. But if 
desired, they may be found thus : When aC becomes contact radius, it has the position a' R 
perpendicular to AE ; but the angle aCP remains unchanged. Therefore make the angle 
a RL equal to it ; then RL is the new position of the generating radius CP, and Z is a point 
on the cycloid. Also, a'L, the instantaneous radius, is normal, and a perpendicular to it 
at L is tangent, to the curve. 

Conversely : Let O be any point on the curve ; about this as a centre describe an arc 
with radius equal to CP. This arc cuts CD, the path of the centre of the rolling circle, in 
S : then OS is the generating radius ; S6', perpendicular to AE, is the contact radius ; and 
d'O is normal to the cycloid. 

82. The Epicycloid. — The describing circle in Fig. 53 rolls upon the outside of another 
whose centre is G. Draw the common tangent at A, set off upon it the length of Aa (any 
convenient fraction of semi circumference AP), and find (56) the arc of the base circle equal 
to that length. Step this off as many times as are necessary upon the base circle, making 
arc AE = semi-circumference AP. The curve is drawn by tangent arcs in the same manner 
as the cycloid. 

The path of the centre C is in this case another circle whose centre is G, and the con- 
tact radii a'R, b'S are prolongations of the radii Ga' , Gb' of the base circle ; which slightly 



40 



MECHANICAL DRAWING. 



modifies the processes of finding the point of the generated curve corresponding to a given 
point of the rolling contact, and the converse. 




Fig. 53. 

83. The Hypocyclold. — Traced, as shown in Fig. 54, by a point in the circumference 
of a circle rolling inside a larger one. The construction is in all respects the same as in 




a' 

Fig. 54. 
the case of the epicycloid, and as the diagrams are lettered to correspond throughout, no 
further explanation is needed. 

In all three of these curves, if the rolling continues beyond E, a new branch springs 
up, which is perfectly symmetrical witli EL. 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



41 




42 



MECHANICAL DRA WING. 



It is particularly to be noted that these branches are tangent to ED, and therefore 
to each other, at E ; and that special care should be taken in constructing those parts of 
the curves which lie in this immediate neighborhood, not only because the change in the 
rate of curvature is here most rapid, but because they are the parts most frequently used 
in practical applications. 

84. Spur-wheels with Epicycloidal Teeth. — Separate diagrams have been given in 
illustration of the epicycloid and hypocycloid, in order to make the construction clear. 
But these may be satisfactorily combined in one exercise, showing the application of these 
curves in forming the teeth of a wheel, as in Fig. 55. In laying out a spur-wheel, the first 
step is to draw t\\& pitch circle, ol which f^fF is a part ; the circumference of this circle is 
then divided into as many equal parts as the wheel is to have teeth, which determines the 
pitch arc AD, embracing a tooth and a space; and the space BD should be a little greater 
than the thickness of the tooth AB : the difference, say ^ to ^V of the pitch, is technically 
called back-lash. 

T\\& face, DG, ox part of the tooth which lies outside the pitch circle, is, as shown, a 
part of the epicycloid APE ; the flank, or part which lies inside, is a portion of the hypo- 
cycloid P'F; and the describing circles for these may be of equal diameters or not, at 
pleasure. But if two wheels are to gear together, the same describing circle must be used 
for the faces of the teeth upon one wheel and the flanks of those on the other, and vice versa. 
The whole height of the tooth, LK, may be from | to |^ of the pitch, LI being greater than 
KI in the proportion of saj' six to five. 

85. It is seen then that in laying out the teeth of wheels only small portions of the 

epicycloids are used ; so that it is not necessary that 
the divisions of the describing circles used in con- 
structing those curves .should be aliquot parts of their 
circumferences. But it is necessary to have equal 
arcs set off on all the circles employed, and this may 
be very readily done as shown in Fig. 56. On the 
tangent at A, set off by the scale, AB — i, BC = 3, 
taking care that AC does not exceed the radius of 

^"^- 56- the smallest circle. 

Then with centre B and radius BC describe an arc cutting the four circles, thus determin- 
ing the four arcs AE, AD, AF, AG; which being each equal to yiC(56), will be equal to each 
other. 

86. The curves forming the outline of one side of a tooth having been accurately con- 
structed, and carried well beyond the limits of the proposed height, the teeth may be very 
expeditiously drawn by the device shown in Fig. 57. A mould or template of the exact 
form of the contour turns freely about a fine sewing-needle fixed in the drawing-board at the 
centre of the wheel. The edges of the teeth being set out on the pitch circle, the template is 
set to coincide with each of these points in succession, until the left-hand sides of all the 
teeth are drawn, when the template is reversed in order to draw the right-hand sides. 

Such a template may be made of thin wood like white holly, to which the contour is 
easily transferred as explained in (71). But one equally serviceable may be made with less 




EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



43 



labor of Bristol-board, upon which the epicycloids may be constructed directly ; the pitch 
circle, and the centre line of the -wheel as well as its centre, should be marked in fine lines, 
after which the working edge of the template may be cut nearly to the line with a sharp 




Fig. 57. 

knife. Then shellac the template on both sides, and when dry it can be worked to the 
precise contour with a file and sand-paper. To use such a template with the pen, gum a bit 
of card to the back, a little away from the edge, so as to keep it slightly raised oi? the surface 
of the paper. 



THE INVOLUTE OF THE CIRCLE. 

87. The Involute is in a sense the converse of the cycloid, being generated by the 
rolling of a tangent line upon a circle. Thus in Fig. 58 the ruler, carrying in the line of its 
edge the pencil P, while rolling around the cylinder describes the curve under consideration. 

It may also be regarded as generated by unwinding a fine inextensible thread from a 
cylinder : the thread being always taut and always tangent to the circumference, its length is 
always equal to that of the arc from which it was unwound. Thus, beginning at O, make the 
tangents BE, AP, DF, respectively equal to the arcs OB, OBA, OBAD, and .so on : then the 
curve passes through the extremities E, P, F, G, etc., of these tangents ; intermediate points 
may be found in like manner. The tangent OG, then, w^ill be equal to the circumference of 
the circle, and if the unwinding be continued, a spiral will be formed ; the distance between 
the successive coils, measured on the tangent to the base circle, as for instance EH, being 
constant and equal to the circumference. 

Considering the curve as traced by the rolling of the ruler : the point of contact, for 
instance ^, is the instantaneous centre (79, 80) ; therefore /i/" is normal to the curve at P, 
and, in general, the normal at any point of the involute is tangent to the base circle, and vice 
versa. 

If the ruler continue to roll in the direction of the arrow, it is clear that after /"reaches 
O, a new branch will be formed as shown by the dotted line ; the two branches being tangent 



44 



MECHANICAL DRAWING. 



to each other, and to the radius CO at its extremity. The method of (80) may be appHed 
also to the involute, which is obviously tangent to arcs described about A, B, D, etc , as 
centres, with radii AP, BE, DF, and so on ; by subdividing the base circle more closely, 




Fig. 58. 

the curve may in this manner be very accurately mapped out, without locating any points 
upon it. 



THE EPITROCHOID. 

88. Although the term epitrochoidal is used in a general sense, including all rolled 
curves, yet custom sanctions also a special sense ; and the curve traced by the rolling of one 
circle upon another, when the marking point is not situated upon the circumference, is the 
one ordinarily meant when "the epitrochoid" is simply mentioned. 

In Fig. 59, if the circle whose centre is C, roll upon the circle whose centre is G, carrying 
a marking point P situated without the circumference, it describes the looped cmve PLE, 
called the curtate epitrochoid. If the tracing point be situated within the circumference 
as at V, the resulting waved curve VWX is called the prolate epitrochoid. Both may be 
drawn by the method of tangent arcs. Since, as in the preceding cases, the rolling circle 
measures itself off upon the base circle, the position of the generating radius can always 
be found as in the construction of the epicycloid, and, its length being constant, points 



exejRcjses in the drawing of non-circular curves. 



45 



on the curve are readily determined. Also, the instantaneous radius being always normal 

to the epitrochoid, the tangent at any point 

can be drawn with the same facility. For 

example, let it be required to draw the tangent 

at L ; with a radius equal to CP, describe 

about L an arc cutting the path of the centre 

C in the point R, and draw RG, cutting the 

base circle in a' : then a'R is the contact radius, 

RListhe generating radius, the instantaneous 

radius a'L is the normal, and a perpendicular 

to it is the required tangent. 

If the circle which carries the marking 
point rolls on the inside of the base circle, the 
resulting curve is called, by way of distinction, 
an internal epitrochoid ; if it rolls upon its 
tangent, the curve is called simply the trochoid ; 
but as the only modifications in the above 
process are those due to the form of the base 
line, no further explanation is necessary. Fig. 59. 




SPIRALS. 



89. A Spiral is a plane curve traced by a marking point moving along a right line, while 
at the same time the right line revolves about one of its points as a fixed centre. 

This fixed centre is called \.\\& pole, and a right line drawn from it to any point of the 
curve is called a radiant, or radius vector. Supposing the angular velocity of the revolution to 
be uniform, the linear velocity of the marking point along the radius vector may be governed 
by any law at pleasure, and thus an infinite variety of spirals may be produced. 

Some of these may also be otherwise generated, as already shown in the case of the 
involute of the circle, in which the tracing point moves uniformly along the tangent, while 
the tangent revolves uniformly about the centre ; but the definition first given is perfectly 
general, and includes them all. 

90. The Archimedean Spiral. — ^The radius vector revolving uniformly, let the tracing 
point travel along it with uniform velocity. The resulting curve is the well-known spiral of 
Archimedes, also called the equable spiral, because the rate of expansion is constant, so that 
the distance between any two consecutive coils, measured on a radiant, is the same. 

If this distance or rate of expansion be given: With it as radius, describe a circle about 
the pole P, Fig. 60, and divide its circumference into any number of equal parts by radial 
lines; divide the given distance PA into the same number of equal parts, and set out from 
the pole P, upon consecutive radii, as I, II, III, IV, distances equal to one, two, three, etc., 
of these subdivisions : the spiral is then drawn through the points thus determined. 

If any two radiants and their included angle be given: Divide the included angle and the 
difference between the radiants into the same number of equal parts ; then set off on the 



46 



MECHANICAL DRAWING. 




EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 



47 



first dividing radial line a distance equal to the least radiant plus one of the subdivisions of 
the difference, and so on; each radiant being greater than the preceding by an amount equal 
to one of these subdivisions. 

To draw a tangent at any point, as O: Drawing a circle about F through (9, and 
rectifying a convenient fraction thereof as in (56), the length of the circumference may be 
found ; and the radial travel per revolution, or rate of expansion, is known. Then, the arrow 
indicating the direction of the revolution, set off on the radiant OP a distance OL, and on a 
perpendicular to it a distance OK, making OK: OL :: circumference through O: rate of 
expansion. Complete the parallelogram, and the diagonal CT'is the required tangent. 

91. Pseudo-spiral. — The above spiral is sometimes used in forming the outlines of cams, 
in mechanical movements, and for such purposes it must be accurately laid out. It is also 
the form assumed by a coiled spring such as the mainspring of a watch or clock. But for the 
purpose of representing such a spring, it is quite needless to make so laborious a construc- 
tion ; an approximation to the true curve is easily drawn as shown in Fig. 6i. The sides of 
the small square being produced, limit the quadrants I, II, III, IV, of which the centres are 
I, 2, 3, 4; tlien the same centres are again used in order, i for quadrant V, 2 for VI, and so 
on : thus the distance between the successive coils is equal to the perimeter of the original 
square. A still closer approximation may be made by using a hexagon, or an octagon, in like 
manner. It is obvious that this more nearly represents the involute of the circle than the 
true equable spiral ; but for the purpose mentioned the curve would not in fact extend very 
near to the pole, in the mechanism itself ; and the accuracy of this method is amply sufficient 
for making the drawings of such springs. 




92. The Logarithmic Spiral.— In this curve the successive radiants which include equal 
angles form a series in geometrical progression, each being greater or less than the precedmg m 



48 MECHANICAL DRAWING. 

a certain constant ratio : in the Archimedean spiral such radiants form a series in arithmetical 
progression, each being greater or less than the preceding by a constant differetice. From the 
above definition it follows that the radiant which bisects the angle between two others is a 
mean proportional between them. Thus in Fig. 62, if the angles APK, KPH, HPC are equal, 
we shall have 

AP : PK :: PK : PH, 

PK : PH :: PH : PC, and so on. 

Had the radiants AP, PH d^nd their included angle, then, been given, the intermediate point 
iT would have been found by bisecting the angle APH, and setting off PK, a mean propor- 
tional between the two given radiants. Had a diameter, as AB, and the pole P, been given, 
the process would have been the same — this angle of 180° is bisected by PC perpendicular to 
AB, and PC is a mean proportional between the segments AP, PB of tlie diameter. 

93. To find the mean proportional between tivo given lines : Set off from any point P on 

any right line a distance PA in one direction equal 
to one of the lines, and in the opposite direction a 
distance PB equal to the other (Fig. 61). Upon AB 
as a diameter describe a semicircle, intersecting at C 
a perpendicular to AB at P: then PC is the mean 
proportional required. 

To find a tJiird proportional to two given lines : 
In Fig. 63 let APh& one of the given lines, and PC, 
perpendicular to it, the other. Draw ^4 (7 and bisect 
it by a perpendicular cutting AP in 0, about which point as a centre describe a semicircle 
through A and C: this will cut AP produced in a point B, and PB is the third proportional 
required ; for by this construction we have AP : PC :: PC : PB. 

This last construction is often useful in extending the spiral ; thus, had the curve ACB 
in Fig. 62 been determined, it is clear that PG, on the prolongation of CP, must be such a 
third proportional to CP and PB. 

94. The length of PG may be otherwise determined ; for since when G lies on the spiral 
we must have AP: PC :: PB : PG, the triangles A PC, BPG will be similar, whence BG must 
be parallel to AC. So also GI is parallel to CB, IL to BG, LQ to GI, and so on. 

This principle is capable of more extended application ; for producing the radiants KP, 
HP to cut the spiral in k, h, it will be apparent that the chords KH, kJi are parallel ; so also 
Bk is parallel to AK, JiG to HC, and so on. 

95. Another Method of setting off divisions in geometrical progression is shown in Fig. 
64. Draw the two lines OM, ON, including any convenient angle, and also AB, cutting them 
both. About O describe the arc AC, and draw CD parallel to AB ; then about O describe the 
arc DE, and draw EF also parallel to AB \ similarly, make OK =■ OB, draw KL parallel to 
AB, and so on as far as may be desired in each direction. 

If now the distances from O to the points of division on either line be set out from the 
pole, on successive radiants including equal angles, the curve thus determined will be a 
logarithmic spiral. 




EXERCISES IX THE DRAWING OE NON-CIRCULAR CURVES. 



49 



This construction may be also adapted to any special case in \vhich t\vo radiants and. 
their included angle are given. Thus, had OA, OB been made equal to those radiants, the 
lengths of others would be found as above; but in constructing the curve, care must be taken 
that the angles between them are made equal to the given angle. 



MECHANICAL METHODS OF PLOTTING CURVES. 

96. Mention has already been made of the use of a slip of paper for locating points upon 
the ellipse (62). Analogous expedients are often available in other cases, enabling the 
draughtsman to work at once more neatly and more rapidl}'. % ^ 
Thus, if the divisions determined as in Fig. 64 be set ofT on 
the radial edge of the strip of paper shown in Fig. 65, it is 
not necessary even to draw the radiants of the logarithmic 
spiral; a circle is described about the pole, and a fine sewing- 
needle, passed through the corresponding point O of the paper 
strip, is fixed in the drawing-board at the centre of the circle, 
whose circumference is first subdivided. Then swineine the 
strip round, setting its edge by each point of division in order, 
the proper point of the spiral is marked with a finely sharp- 
ened pencil. By making the divisions of the radial edge 
equal, the Archimedean spiral can be laid out with the same 
facility. 

97. In Fig. 66, the lower edge of the paper strip is tan- 
gent to the circumference of the circle, and the subdivisions 
upon it are equal in length to the corresponding subdivisions 
of that circumference. Turning this as before about a needle 
fixed at the centre O, setting the radial edge OR by the successive points on the circum- 
ference, and marking with a fine pencil-point the corresponding divisions of the lower edge 
(counting from zero to the left when the rotation is in the direction of the arrow), the curve 




Fig. 64. 




Fig. 65. 

thus determined is the involute of the circle ABC. By counting from zero to the right, the 
direction of the rotation being unchanged, points are determined in the curve ADE, the com- 
panion to the involute. Of these two curves, the former is traced by unwinding an inexten- 



5° 



MECHANICAL DRAWING. 



sible fine thread from a fixed Q.y\\x\6.&x\ the latter by unwinding it from a cylinder which turns 
upon its axis so that the thread is always tangent to the circumference at the same point. 




Fig. 66. 

But the above simple device shows that they may also be defined as the paths of points 
travelling uniformly along a right line, while the line itself revolves uniformly about a fixed 
point not situated upon that line. 

These, however, are special cases, the travel along the line during each revolution being 
equal to the circumference of the circle to which the line is tangent. But it need not be ; 
and by means of different graduations of the paper strip it is easy to plot, without drawing a 
single construction line except the divided circumference, this whole class of spirals, of which 
the involute and its companion are the most conspicuous representatives. 

98. The path of any point on the connecting-rod of a direct-acting engine may be plotted 
as easily as the ellipse, the only difference being that one of the fixed points on the paper 
slip is to be kept always on the circumference of the circular path of the crank-pin. If the 
connecting-rod is coupled to two equal cranks, like the side rod of a locomotive, and these 
cranks turn in opposite directions, the middle point of the rod traces a lemniscate ; which 
may also be easily plotted by the above method. 

The piston-rod of an oscillating engine passes always through a fixed point, which may 
be represented by a fine sewing-needle stuck into the drawing-board : against this let the edge 
of a straight slip of paper be pressed, a fixed point thereon being kept on the circumference 
of the path of the crank-pin, and the curve described by any point on the rod can be plotted ; 
if this circle be of infinite radius, the curve becomes the conchoid, while if the centre of 
oscillation be located on the circumference, it becomes the lima^on. 

99. Any of these paths may be constructed at pleasure as exercises in the drawing of 
curves ; and as a final illustration of this method of manipulation, we give in Fig. 67 three of 
the curves which may be plotted by means of Newton's Square. 

This was devised by that renowned geometer for drawing the cissoid ; of which the 
generating circle is that whose centre is O in the figure. On the vertical line through O a 
needle is fixed at E, the distance OE being equal to the diameter of the circle. Against the 
needle the side BE of the square DDE is kept, while the point A, whose distance from B is 
equal to OE, is kept always on the horizontal line through 0. Then the middle point P oi 



EXERCISES IN THE DRAWING OF NON-CIRCULAR CURVES. 5I 

AB traces the cissoid PC\ a point D, so situated that AD = AP, traces the Avaved curve 
called the wizard, while the corner B traces the looped curve called the sprite. 

It will be clear that when P reaches C, the working edge P)B will coincide with the 




Fig. 67. 

vertical line OE: the square must then be turned over, and the under side used for continuing 
the curves to the right. 



100. Before proceeding farther, we give on page 52 a number of examples of " corner- 
pieces," suitable for borders of a more ornate style than the simple ones heretofore employed. 

From these the ambitious student may select such as suit his fancy or his estimate of his 
skill, — with the understanding that they are to be executed with the same care and subjected 
to the same criticism as any part of the exercises contained within them. With this under- 
standing, no instructor possessed of ordinary common-sense will attempt to control his 
selection, except by hints and suggestions. He may and very likely will at first set his heart 
upon something glaringly inappropriate to the case in hand ; but he will be much m.ore readily 
convinced of this after he has drawn it than before. And good taste in this can no more be 
acquired without experience, than can perfection in the execution of the exercises themselves. 



52 



MECHANICAL jpHAWJNG. 




J^ 



T 



I I II 






53 



CHAPTER III. 
THE PRINCIPLES OF PROJECTION. 

101. Pictorial representations, such as photographs and perspective drawings, while they 
may give perfectly correct ideas of the form, proportions, and general appearance of the 
objects depicted, are of no use for constructive purposes. Parallel lines are represented by 
converging ones, equal lines by unequal ones, right angles by those which are obtuse or 
acute : so that, in general, no accurate information can be obtained by direct measurement. 

Working drawings, however, are needed for the absolute guidance of the workman, 
who by their aid alone must be enabled to produce precisely what is intended, and that only. 
Such drawings, then, must exhibit the exact forms and dimensions, and also the true relations 
of the various parts, in a manner both direct and unmistakable. 

The difference between a perspective and a working drawing is sometimes expressed b)' 
saying that the former shows things as they seem, the latter as they a7-e: but the precise 
nature of this difference may be explained as follows: 

102. Suppose the eye (regarded as a single point) to be placed in a given position in 
front of a plate of glass, and to be looking perpendicularly against its surface, which is called 
the picture plane. Any point of an object beyond the glass is visible, because light is re 
fleeted (or projected) from that point Lo the eye in a right line, called a visual ray. And 
the point in which that ray pierces the picture plane is the representation, or projection, of 
the point from which it came. If a sufificient number of such points be found, the outlines of 
any object may be fully determined, and the drawing thus made will present to the eye, if 
placed in the position originally assigned to it, the same appearance as the actual contour of 
the object itself. 

103. When the eye is placed at a finite distance from the picture plane, the drawing is 
said to be in perspective. In this case the visual rays converge, and but one of them can 
coincide with the axis of vision, which is perpendicular to the plane. Nevertheless, the eye, 
without straining and without turning, can receive distinct impressions from all visual rays 
embraced in a cone whose angle at the vertex is about sixt}^ degrees : the picture plane cuts 
from this cone a circle, within which the drawing should be included. 

104. Other things being equal, the visual rays will converge the less rapidly the farther 
the eye is removed from the picture plane. If the eye be placed at an infinite distance, the 
rays will become parallel to each other and to the axis of vision, which is still supposed to be 
perpendicular to the plane. In these circumstances the drawing is called an orthographic 
projection, the visual rays are usually designated as projecting lines, and the picture plane is 
called the plane of projection. It will readily be seen that a plane may be so interposed as 
to cut the parallel projecting lines at any other angle ; that is to say, the eye, although 
infinitely remote, may not be directed perpendicularly toward the plane: the drawing is then 



.'54 



MECHANICAL DRAWING. 



called an oblique projection, and in some cases is of great utility and convenience. The 
former arrangement, however, is much more extensively used, and thorough familiarity with 
it is necessary to an intelligent use of the latter. For the present, then, it will be understood 
that in speaking of projections reference is made only to those constructed on the ortho- 
graphic system. 

Q R 




Fig. 68. 

105. Fig. 68 is a perspective drawing of a rectangular prism, placed within a case formed 
of plates of glass, upon which the projections of tiie object are made in the manner above 
explained : thus, aa is the projection of the front face A, bb is that of the side face B, and cc 
is that of the upper face C. The left side face of the prism is not visible, but its projection 
upon the left side of the glass case is shown at dd. 




Fig. 69. 

Now suppose the sides and the top of the case to turn outward, about the lines MN, OP, 
and MO, until they coincide with the front plate NO. The four projections will then be 
arranged as shown in Fig. 69, the dotted lines representing the sides of the glass case. Re- 



THE PRINCIPLES OF PROJECTION. 55 

garding A as the front of the object, the projection a is commonly called a front view, b and d 
are called side views, and c is called a top view ; although in the case of an object lying upon 
its longest side a may also be properly called a side view, b and d being then spoken of as end 
views. These three projections, which are made upon vertical planes, are also, particularly in 
relation to architectural subjects, designated as elevations ; c being in that case called a plan 
view, or simply a plan. 

106. In reference to the arrangement, it is to be specially noted that b, which is a view 
of the object /rw« the right, is placed at the right of the front view a, while d, a x'lcw frotn the 
left, is placed at the left. This may seem of small consequence, and in fact it is so in repre- 
senting so simple an object ; but suppose A to be for example a side view of a locomotive 
heading to the right, and it will readily be seen that the view of the headlight and the cow- 
catcher, which is only to be had by looking at the engine from the right, is most naturally and 
appropriately placed at the right side of the front view ; as the parts seen in that end view 
are thus brought nearer to their representations in the first one. It may be said that without 
any reference to planes of projection, this arrangement of the views is that which would be 
suggested by simple common-sense to any one who, provided with paper and pencil, should 
sketch an object as he sees it, first from directly in front, and then from the right or the left, 
as the case may be. 

107. In regard to the plan, or top vievv, however, there is this to be said, that no am- 
biguity or doubt can arise if it be placed below the elevation, or front view. Because it is 
invariably a view of the object as seen from above. Under no circumstances should a pro- 
jection be made which involves the idea of the eye being placed beneath the object and look- 
ing upward, — with the possible exception of a design for a frescoed ceiling. Such projections 
are occasionally made, and are even recommended b}- some ; but that does not lessen the 
force of the objection, that they are difficult to read because they represent things as seen 
from a most unusual direction. It is certainly often necessary to know the appearance of the 
lower end of a part of a machine, for example, which is to be placed in a vertical position. 
But in order to find it out, one does not hold the object over his head and look upward : the 
natural proceeding is either to turn it upside down, or to lay it on its side and look at it 
horizontally ; and the same course may be adopted in making the projections, — of which more 
hereafter. 

Consequently, a top view may be placed either above or below the elevation, as may 
appear most convenient or suitable ; in the case of the locomotive, for instance, it is easy to 
see that the top view, being of nearly uniform breadth, will present a more satisfactory ap- 
pearance if placed under the side view, in which usually the rail is represented, than if put so 
far above as to be well clear of the smoke-stack, sand-box, cab, and steam-drum, which form a 
very irregular contour. 

108. In Fig. ,'0, A, B, and C are the projections of the three faces of the prism corre- 
spondingly lettered in Fig. 68. It is to be noted that each projection exhibits only two 
dimensions ; thus the front view gives the height and the breadth, the side view gives the 
height and the thickness, the top view gives the breadth and the thickness. Also, that the 
height ad is the same in A and B, the breadth ab is the same in A and C, while the thickness 
ac is the same in B and C. 



56 



MECHANICAL DRAWING. 



It will also be observed that while the perspective representation conveys in one view a 
correct and complete idea of the form of the object, no one, nor yet either two, of the pro- 
jections are suf^cient to do this. Were A and B only given, it could not be told whether any 
or all of the vertical edges were square or rounded off ; and if A and C were the only views, 
the top or the bottom, or both, might be either plane, or curved transversely ; but the three 
together define the form completely. 




TT 




109. Again, it will be noted that each of the projections represents a face of the prism in 
its true form and size. A very little study of Fig. 68 will show that this is because the planes 
of projection are parallel to those faces; the projection of a line is made by connecting the 
projections of all its points, from which it follows that if a line be parallel to the plane of 
projection, it will be represented on that plane by a similar and equal line. For example, had 
the object been an upright cylinder, the projections a, h, and d would still have been rectan- 
gles, but c would have been a circle. The position of the object is of course arbitrary; thus, 
the prism might have been turned part way round, or placed upon an edge or a corner. But 
the position actually shown is the simplest one, and the one which would naturally be selected 
for making a working drawing, which Fig. 70 is properly called, since by its aid the object 
represented can be made by a workman, who takes his measurements directly from the pro- 
jections. 

The prism is shown as resting upon the horizontal plane NS, forming the bottom of the 
glass case. This in Fig. 70 is indicated by the horizontal line NN, called a base line ; which, 
though not essential, is in many cases introduced with excellent effect, for it at once shows 
unmistakably that the view beneath which it appears is an elevation, and not a plan. 

110. This position of the prism, which may be also described as that in which its lines 
and faces are parallel and perpendicular to the paper in each of the projections shown in 
Figs. 69 and 70, has been selected not only by reason of its simplicity, but because, as will be 
shown presently, it is the starting point from which we set out in determining the projections 
of the same object in oblique positions. In those positions the projections, though still 
orthographic, may not show the lines and angles in their true dimensions ; and in order to 



THE PRINCIPLES OF PROJECTION. 



57 



construct them correctly, it is necessary to possess the information conveyed by Fig. 70, the 
construction of which is indeed an essential preliminary. 



m 



ta 



m. 



-n 




Fig. 71. 

Before going on to discuss those more complicated matters, however, it may be remarked 
that in a working drawing the most simple and direct representation is always the best ; 
so that the principles already explained are sufficient for practical application, to very good 
and useful purpose, in the delineation of minor mechanical details : of which accordingly we 
introduce one or two examples. 

111. Fig. 71 represents a horizontal C}'linder, having at one end a circular flange, and at 
the other a larger square one, in which are four bolt-holes. 

In this case two views are sufficient, as a top view would be identical Avith the side view. 
The circular flange on the right-hand end being the smaller, a view from that end defines the 
whole, as the square one at the left is visible beyond, as also are the bolt-holes. Parts not 
visible are indicated by dotted lines, as for example d in the front view and c in the end view, 
and the outlines of the bolt-holes in the former. 

In Fig. 70 it will be noticed that the projecting lines aa, bb are represented, in dotted 
lines, for the purpose of indicating the relations between the different projections. In illus- 
trative diagrams, this is necessary in order to aid the reader in following the explanations. 
But when the subject is mastered they are no longer required, and in working plans are very 
rarely introduced, the practised eye being able to read the drawings, and compare one view 
with the others, without such assistance. 

112. But when, as in Fig. 71, the object is symmetrical about an axis, and especially if it 
be a surface of revolution, it is customary to connect such views as these by a continuous 
centre-line, as lih in the figure ; and in the end view to locate the centre by a transverse 
centre-line, vi\ Similarly, the centres of the bolt-holes are marked in the end view by short 
vertical and horizontal centre-lines, corresponding horizontal ones, mm, jm, being drawn in 
the side view. These are of the first importance, since it is by measurements from these lines 



58 



MECHANICAL DRAWING. 



that the mechanic lays out his work, and they should never be omitted. Being imaginary 
lines, they should be drawn as fine as possible, and should never terminate in a bounding' 
line of the object represented, or of any part of it. In drawings of machinery, if different 
colors are used, it is customary to draw them in red. 

c o 




Fig. 72. 

113. Fig. 72 represents a small bed-plate, with four facing-strips and lugs on its upper 
face. Though not absolutely necessary in so simple a subject, the views A and B are shown 
in section, as showing more distinctly the box-like form of the bed-plate : such views are 
commonly called sectional elevations. 

In this instance three views are necessary ; and the figure shows an arrangement of them 
which is frequently adopted in order to economize space. The breadth being considerable as 
compared to the height, it is clear that if, as in Fig. 70, B had been placed at the right of A, 
the three views would have occupied more space than they now do. But when it is placed 
at the right of the plan, as in the figure, it will be at once seen that C is as correctly a top 
view of B as it is of A. 

When this arrangement is adopted, the fact that B is an elevation should be emphasized, 
by the proper use of shadow-lines according to the direction of the light shown by the arrow, 
as well as by the manner in which the dimensions, if any, are figured ; so that it shall always 
be looked at as it should be, that is, by turning the paper a quarter round to the right, thus 
changing horizontal to vertical, and vice versa. If as in this example the subject is such as to 
render the introduction of a base-line appropriate, it forms a most effective aid to the correct 
reading of the drawing. 

114. Figs. 70 and 72 are similarly lettered, and by comparing them a clearer idea may 



THE PRINCIPLES OF PROJECTION. 59 

perhaps be gained of the correspondence of dimensions in each two of the three projections. 
It is now time to impress upon the reader that advantage should be taken of this correspond- 
ence in constructing the drawings. Thus, for instance, the lines ba in the top view C, and ad 
in the end view B, should be drawn at one setting of the T-square ; also ac in the top view 
and ad'xw the front view A, should be drawn at one setting of the triangle, and so on. 

This is in accordance with the principle previously enunciated, that as much as may be 
should always be done with one adjustment of an instrument. But a great additional advan- 
tage of thus setting up the different views simultaneously as far as possible, lies in the fact 
that errors, to which even the most experienced are liable, are thus more likely to be at once 
detected than if each view were constructed separately. This is of special importance in 
planning, aside from the more perfect conception of the proposed structure gained by aid of 
the different views ; but the saving in time is also considerable, and the habit should be 
formed at the start and always retained. 

115. And so should the habit of working from centre-lines, which in general should be 
among the first lines pencilled in. For example, the lines JUi and vv should be first laid down 
in drawing Fig. 71 ; equal distances above and below hh are laid off to locate the centre-lines 
mm and «« of the bolt-holes in the left-hand view, and the half diameter of each bolt-hole is 
set off by the scale on each side of its own centre-line ; nothing could be less workmanlike, or 
more likely to lead to errors sooner or later, than to construct such drawings by measuring 
from the edge of a flange to the side of a bolt-hole, and from the side of one hole to the side 
of another. In short, the draughtsman should lay out his work on paper very much as the 
mechanic lays out his on metal. 

116. In Fig. 73 are given three views of a piece formed by milling the upper part of a 
vertical cylinder into the form of a hexagonal prism. The top view is first made by drawing 
a circle of a diameter equal to that of the cylinder, and inscribing within it a hexagon. The 
points a, b, c, etc., represent the vertical edges of the prism, which in accordance with what 
precedes, will in the front view appear perpendicularly under these points, and be seen in their 
true lengths, as al, ck, etc. The edges al, dg coincide with the visible outlines of the cylinder, 
Ip, gh, which also are represented in the top view by the points a, d. The upper front edge 
be being horizontal, and parallel to the paper in the front view, appears of its true length in 
both the views A and C. But ab, cd, though horizontal and consequently seen in their true 
length in C, are inclined to the paper in the front view, where they appear what is technically 
called foreshortened, that is, of less than their actual length. 

A third projection of so simple a thing is not necessary; but if as an exercise it be 
required to make it, we have now all the data : the altitudes will be the same in ^ as in ^4, the 
diameter x of the cylindrical part is also of course the same, and the "short diameter" y of 
the hexagonal part is given in C, and will be equal to that in the new view ; to which it is 
transferred by setting off hy on each side of the vertical centre-line. 

117. Foreshortening, then, means the apparent reduction in the length of a line caused by 
viewing it obliquely. It was already understood that a line parallel to the paper is projected 
in its true length, while one perpendicular to it appears as a point. In intermediate positions 
it must therefore appear of intermediate lengths, and from Fig. 73 it will be clear that the 
amount of foreshortening depends upon the angle which the line makes with the paper, being 



6o 



MECHANICAL DRAWING. 



the greater as that angle approaches 90°. For example, the line reappears shorter in the view 
A, where the angle is 60°, than in the view B, where it is 30°. 

A little study also shows that in the former view the lines ab and cd, and in the latter the 
lines cd and de, are equally foreshortened, because they make equal angles with the paper. 




Fig. 73. 

From this follows the deduction, which in subsequent operations will be found of great 
importance and utility, that lines which are equal and parallel in space will appear equal 
and parallel in any projection ; — a fact which the student should fix in his mind so that it 
will stay there. 

118. It will have been inferred that if two projections are sufificient to represent the 
object completely, — that is, so that by their aid a workman can make it and cannot make 
anything else, — no more are necessary. Ordinarily, two are not enough, while three are — but 
not by any means in all cases; and when they are not, as many more must be added as the 
occasion demands. 

Fig. 74 represents a rectangular prism with projecting pieces upon its difTerent faces, 
purposely so formed and located that they cannot, even by dotted lines, be so defined by 
means of the three views thus far considered, as to convey clear and unmistakable ideas of 
what is intended. In addition to the views A, B, and C, a view D of the left face is placed at 
the left side of the front view A. But a back view is also necessary ; and it may, in conformity 
with the foregoing methods, be logically placed as shown at E. For regarding B for the 



THE PRINCIPLES OF FROJECTION. 



6i 



EH 



T \ ^ 

..J \ 

-1 \ 

■ I I 







i, ^~^ 



-□ 



62 MECHANICAL DRAWING. 

moment as a new front view, it will be clear that A will be a view of the object from the left, 
and E a view of it from the right. 

119. Again, it sometimes happens that although a certain number of views are sufificient 
to define the construction, and that too without ambiguity when once they have been studied 
out, yet there is difficulty in reading them. ' If this difficulty can be lessened, and the meaning 
more clearly expressed, by the addition of other views, it is the draughtsman's duty to add 
them. We are entering upon the study of projections for a definite practical purpose, that is, 
the making of working drawings for workmen to use. They cannot use the drawino-s if 
they cannot read them ; and if they cannot read them easily, then in a practical sense the 
drawings are bad in proportion to their obscurity, no matter how accurate or how perfect in 
execution. 



CHAPTER IV. 

PROJECTIONS CONTINUED.— OBJECTS IN INCLINED POSITIONS. 

120. In Fig. 75 a rectangular prism is shown, as before, in its simplest position. Using 
a geographical illustration, A is a. view from the south, D a view from the west : let it now be 
required to make a view from the south-west, as indicated by the arrow f/i. It will be 
readily seen that the horizontal plane on which the prism stands will be represented by a 
new base-line N'N', perpendicular to the arrow ; and that the vertical edges of the prism will 
be represented by lines ad, be, etc., drawn through the points a, b, etc., of the view C, parallel 
to the arrow : since the altitude is not affected by thus changing the point of view, these 
edges still appear in this new front view, A', of their true length, just as they did in the 
original front view A, 

121. This is as though the glass case in Fig. 68 had been turned part way round to the 
right, while the prism within it retained its original position. The same result may be 
reached by supposing the case to remain stationary, while the prism, still standing upright 
on the bottom of the case, is turned to the left through the same angle. This is shown in 
Fig. 76: it needs no argument to show that the upper face of the prism will undergo no 
change of form or size, as seen from above, by this rotation about a vertical axis, so that the 
top view in this figure is simply a copy of C in Fig. 75. Also, the base-line XN remains the 
same, as well as the altitude of the prism, so that the front view A' is constructed by drawing 
the vertical edges ad, be perpendicularly over the points a, b, which represent them in the 
new top view, and similarly for the other edges ; the heights being the same as in views 
A, D, of Fig. 75 ; and the views marked A' in Figs. 75 and ^6, clearly, are identical. 

122. Now let it be required to construct a view of the prism in its new position, Fig. y6, 
as seen from the right ; or, geographically, from the east, as indicated by the arrow k. It 
may be, perhaps, more clearly seen how this is done, by at first adopting the arrangement of 
views employed in Fig. 72. The horizontal plane will be represented by a new base-line 
N"N", perpendicular to the arrow ; turning the paper round so as to make this line hori- 
zontal, it will be obvious that the vertical edges will be represented by ad, be, etc., drawn 
through the points a, b, etc., of the top view, parallel to the arrow k, and of the same length 
as in the front view A' : which is in fact their true length, because they are parallel to the 
paper in this view also. 

If this new view B" be turned a quarter round, and placed at the right of A', it will 
appear as B' ; which is the arrangement most frequently adopted. 

123. The student should now be able without difficulty to perceive that the view B' may 
be at once constructed, without the use of B" , as follows: Draw in the top view a line //, 
parallel to NN: this represents the edge of a vertical plane, in this instance containing the 
front vertical edge of the prism ; as though the object had been placed in contact with the 
front plate of the glass case in Fig. 68. To an observer placed at the east and looking toward 



64 



MECHANICAL DRAWING. 



PU \A 




bo 

•H 




Tti 5) 



ct 









^ 


/^ 




o 


M'-J ^\. 


\ 


^^ 






OBJECTS IN INCLINED POSITIONS. 65 

the west, as shown by the arrow k, this plane will also present its edge, and appear as the 
vertical line// in the view B' . To such an observer, also, the edge represented by b in the 
top view will appear at the extreme left, and coincident with// ; while the edge represented 
by c will appear at the extreme right, and at a distance from // equal to cs\ to which there- 
fore be in the required view B' is made equal. Similarly, the perpendicular distance of the 
edo-e (7i/from the plane// is ar in the top view ; and as ar is seen in its true length, whether 
we look down upon it vertically from above, or look at it horizontally from the right of the 
object, this length is set off as ba in the new view. 

124. This last mode of procedure, evidently, would be decidedly the preferable one were 
it desired to supplement the front view A' in Fig. 75, by a side view as seen from the direc- 
tion indicated by the arrow k. This has been done in the illustration, and it is now perfectly 
obvious that B' is the same in Figs. 75 and 76. In short, the only difference between these 
figures consists in the fact that in the former the original view C is made to do double duty, 
being also the top view in the group C, A', B' , in reading which the paper should be so held 
as to make the new base-Une IST JSF horizontal. 

Virtually, then, in Fig. 75, and actually in Fig. "j^, the original position of the object has 
been changed by rotation about a vertical axis, that is, one perpendicular to the paper in the 
top view ; and it is to be noted, that not only is tJiat top view Jinelianged, but so also are the 
altitudes, or measurements parallel to the axis: and it is by reason and by means of these two 
circumstances that we are enabled to construct the projections of the object in its new 
position. 

125. Now, referring to Fig. 68, we may suppose the prism to be tilted for^vard or back- 
ward, rotating about the front or the back edge of its lower base, or in other words, about an 
axis perpendicular to the paper in the side view. In this case it will readily be seen that the 
side view of the face B will not be changed, nor yet the apparent breadth of the front face A, 
the projection of which in the new position will still be bounded by the same vertical lines 

a, a. 

In Fig. yy, therefore, supposing A and B to be the front and side views of the same prism 
as in the preceding figures, we first construct B' . a copy of B : the edges a, b, c, d are still 
horizontal, and are seen in the new front view A' in their true length, and at the same 
distances above the new base-line N' N' as in the side view: so that A' is readily constructed 
from A and B' by means of the T-square and triangles only. Next suppose a vertical refer- 
ence plane, //, to be passed through the edge a: then in the top view C the other edges 

b, c, d will appear at distances from// respectively equal to bt, es, dr in the side view B'. 

Instead of copying the original side view B, we might have drawn a new base line, N"N"\ 
then looking down perpendicularly toward this new horizontal plane, as shown by the arrow 
k, the edges a, b, etc., would have appeared of their true length, which is given in A : and 
thus the top view C" is constructed, the process being exactly analogous to that adopted in 
Fig- 75- It is hardly necessary to point out that this result is precisely the same as if the top 
view C had been laid out under the side view B', by projecting the edges a, b, etc., vertically 
downward, and measuring their lengths from A'. 

126. Similarly, we may imagine the prism in Fig. 68 to be inclined to the right or the 
left, by turning about an axis perpendicular to the front plate of the glass case. The front 



66 



MECHANICAL DRAWING. 



view will then evidently undergo no change in form or size, nor will the thickness, as seen in 
the side and top views. The construction of the drawing in this new position is so similar to 




c 



ri^.77. 



that in the preceding case that no further explanation is necessary ; and indeed Fig. "JJ illus- 
trates it perfectly, if we regard j5 as a front view and y^ as a side view. 

127. In each case, then, the prism is changed from its original position in Fig. 68, by 



OBJECTS IN INCLINED POSITIONS. 67 

rotating it about one of its edges as an axis ; in each case the view in the direction of that 
axis, and also all measurements parallel to it, remain unchanged. Clearly, the same would 
hold true, whatever the object represented, and the rectangular block was selected only on 
account of simplicity. But before going on to exercises of greater complexity, it is to be 
noted that no one of the changes thus far made is sufficient to meet all possible require- 
ments ; but by combining them the prism may be drawn in any position which can be as- 
signed. 

128. In Fig. 78 the same object is represented in the position described in (126). We 
may now regard this as the original position, and treat it in the same manner as before. In 
Fig. 79, for example, is shown the result of next turning it about a vertical axis ; the top 
view, C , being therefore a copy of C in Fig. 78, but in a different position. Since the heights 
also remain unchanged, the new front view A' is constructed by projecting the points a, b, c, 
d, etc., vertically upward from C , with the triangle, and horizontally across from Fig. 78 with 
the T-square. 

In the side view B', which is from the right, as indicated by the arrow k, the heights are 
also the same as in ^' ; and this view is constructed precisely as was the corresponding view 
in Fig. 76, by measuring the distance of each point from a vertical reference plane//; which 
for convenience is passed through the extreme left-hand corner d ; thus, for instance, in both 
C and B' the distances as, ft, etc., are of the same magnitude. 

129. It need hardly be remarked, that instead of copying the view C from Fig. 78, we 
might have adopted the course illustrated in Fig. 75. Thus, drawing a new base-line N"N", 
project the various points a, b, c, etc., perpendicularly toward it : the height of each point 
above the new base-line is the same as that above the original base-line in the front and side 
views A and B, and is transferred from one of them by direct measurement. In this way the 
view A' in Fig. 78 is constructed : it corresponds to the view A' in Fig. 79, and would have 
been identical with it had the direction of the arrow in been determined with a view to such 
coincidence, as was done purposely in Figs. 75 and "j^. 

No new principle or mode of operation, then, has been introduced, but the object has 
been turned about two axes in succession. After the revolution about the second axis, it is to 
be observed that in two of the three views, the foreshortened representations of the faces of 
the prism are no longer rectangular, some of the right angles appearing obtuse and others 
acute. But each face is nevertheless represented by a parallelogram, in accordance with the 
deduction previously made, that all lines which in space are actually parallel will appear 
parallel in any projection ; and if also equal in space, their projections will be equal. 

This should be always kept in mind, not only because it serves to lessen the labor of 
construction, but because it is most important as a test of accuracy. And the triangles 
should always be used as parallel rulers in drawing the inclined sides ; for since no meas- 
urements are to be made from these foreshortened views, it is preferable, if there is any error 
at all, that it should be in the magnitude, and not in the form. 

130. We may now proceed to rotate the prism about the third axis, which is perpen- 
dicular to the paper in the side view. The form of this view is not changed by this rotation, 
and B" in Fig. 80 is a copy, in a new position, of B' in Fig. 79. In making this copy, it will 
be found more accurate and more expeditious to construct first the rectangle syxt within 



68 



MECHANICAL DRAWING. 



riff.so. 




OBJECTS IN INCLINED POSITIONS. 69 

which B' is enclosed, and then to set off along its sides the distances sd, sa, by, etc., thus 
avoiding the direct construction of acute or obtuse angles. 

The front view A" is next laid out by projecting each point vertically upward from the 
view A' of Fig. 79, and horizontally across from the side view B" . In the top view C" each 
point is of course vertically over its representation in A", and its distance from any con- 
venient reference plane pp is the same as in the side view: thus as', ft', etc., are equal in 
these views, — precisely as in Fig. 79. 

As in the cases previously considered, the copying of the side view might have been 
avoided, and B' made to serve as a member of the group in the final position by drawing a 
new base-line, and transferring from A' the measurements of breadth for the new front view; 
but in this case with less advantage, if indeed with any, in respect to the saving of labor; for 
which reason the process has not been illustrated. 

131. Of course, by rotating the prism about the^axes in different order, or in different 
directions and through different angles, it may be placed in other positions than those shown 
in these diagrams: the student should now be able to construct these for himself, and will 
find it a profitable exercise to do so. 

Since there are at present but three dimensions in space (and there is no immediate 
prospect of an increase in the number), the prism shown in Fig. 81 may by successive 
rotations about the three axes on, ob, oc be placed in any position whatever. 

The operations which have been explained, then, cover the whole ground, so far as this 
object is concerned : given at the outset its projections in the simplest position, we are now 
able to represent it correctly in the most complicated position, in which none of its sides 
appears of its true length, and none of its angles is shown in its actual magnitude. 

In dealing with objects of any other form, the same principles hold true, and the same 
modes of procedure are employed. But there is a means of abbreviating the work when it is 
required to represent an object in such a position as that shown in Fig. 80; and in explaining 
this it may be as well to adhere to the simple prism for illustration. 

132. The position alluded to was derived from the one shown in Fig. 68, by successive 
rotations about axes parallel to three adjacent edges of the prism. And the proposed 
abbreviation depends upon the fact that the successive rotations about any two of these 
are equivalent to a single rotation about another axis lying in the plane of those two, and 
passing through their intersection. This will be easily seen by the aid of Fig. 81, in which 
oa, ob lie in the horizontal plane of the base. 

A rotation about oa will raise the point y above that plane ; a subsequent rotation about 
ob will produce the same effect upon the point x, while the point o will remain fixed. It is 
clear that a single rotation about any horizontal line vin passing through will raise both 
X and y above the plane, while o as before will remain stationary. The more nearly vin 
coincides with ob, the higher will x be raised by a rotation through a given angle, and the 
more nearly it coincides with oa, the higher will y be raised. By properly selecting the 
direction of mn, therefore, it follows that both x and y may be lifted to any given distances 
above the horizontal plane ; this will of course determine the inclination to that plane of the 
new position of oiv, which line will remain in a vertical plane perpendicular to mn. The 



7° 



MECHANICAL DRAWING. 




direction of mn and of the rotation about it being arbitrary, any possible position can be 
given to the prism by an additional rotation about the vertical axis oc. 

133. It will often happen in practice that the inclination of the base will be given by 
requiring that the points x and y shall be higher 
than by certain given distances ; and the first 
thing is to determine the direction of inn which 
will effect this, which may be done as in Fig. 
82. It is clear that the distances through which 
those points will rise are directly proportional ^ 
to their distances, xd and yg, from the axis mn. 
Let these be, for example, in the ratio of three 
to five: divide oy into five equal parts, and set 
off ol equal to three of those parts ; draw xl, 
then nion parallel to xl is the axis required. 
For, drawing xd and _;/^ perpendicular to ntn, and 

producing xl to k, we have 

Vg '.kg or xd :: yo : lo. 

134. Now let it be required to draw the projections of the prism shown in Fig. 83, after 
it has been turned first about inn (which will cause the vertical edge^ represented by b in the 
top view C to appear in the direction bw perpendicular to /;/«), and subsequently around a 
vertical line at b, through the angle ivbz. 

The first step is to construct, as in Fig. 75, the view A' , looking in the direction mn. 
Then, supposing that the height to which the corner e, for instance, is to be raised had been 
assigned : describe about e in this view A' an arc with radius equal to that height, and draw 
the base-line N' N' through a, tangent to that arc. 

Next construct a top view C" , looking down perpendicularly to the new horizontal plane 
thus indicated. In this view the edges ^/, ^^, etc., will appear parallel to N'N', and their 
distances from any convenient vertical reference plane // will be the same as in the original 
top view C. This new view C" evidently represents the object with the required inclination 
to the horizontal plane : now make the angle z"bv" equal to the angle zbv, shown in connec- 
tion with the view C ; then v"b will be the direction in which we must look to make the front 
view in accordance with the assigned conditions. 

135. Draw then a final base-line N" N" perpendicular to v"b\ project each point in C" 
perpendicularly toward this, the horizontal plane, and make the height of each point above 
N" N" equal to its perpendicular distance in A' above N' N' : this new front view A" is the 
one required. From A" and C", a side view B ' is now readily made by the aid of a vertical 
reference plane />'/>', the distance of each point from this plane being the same in the side 
view as in the top view C". 

Now, holding the paper so that N" N" is horizontal, the three projections are seen to 
represent correctly the prism in the position called for ; and as there is but one intermediate 
view to make in passing from the simplest to the most complicated position, the labor of 
construction is reduced to a minimum. 



OBJECTS IN INCLINED POSITIONS 



N 




>r 



rio-.ss. 




72 MECHANICAL DRAWING. 

136. It will probably be most often the case that such views are required to be so placed 
upon the paper that the base-line shall be horizontal : usually, also, there would be no reason 
for keeping a record of the process of construction, and therefore no occasion for inking in 
the intermediate view A'. Nor is there any necessity for even pencilling it upon the paper; 
which may be avoided by constructing it upon a small piece of tracing-paper, on which should 
also be traced the original top view C, with the line//. Next drawing a line mi, making with 
N' N' an angle equal to vbz, the tracing-paper may be turned around and so adjusted as to 
make tm vertical ; after which C" is drawn at once upon the sheet, below the edge of the 
tracing-paper, which is removed after the other views have been constructed. 

137. It should now be clear that this ultimate position of the object might have been 
defined by giving the angle which a line, originally vertical, shall make with the horizontal 
plane, and also the position of the vertical plane in which that line shall lie ; for in Fig. 83 
the former is determined by the first rotation about mn, and the latter by the second rotation 
about the vertical axis ab in the front view A. 

Thus in Fig. 84 the vertical axis oa appears in the top view C as the point : let it be 
required to draw the pyramid when this axis lies in the plane ow, and makes a given angle 
with the horizontal plane. 

First construct the view .^', looking perpendicularly toward the plane ow; in this view 
the base of the pyramid will appear as the straight line bd perpendicular to oa, and N'N', 
making with oa the given angle ocb, will represent the horizontal plane. From A' and C the 
new top view C" is constructed as in Fig. 83 ; make the angle w"o"v" equal to the angle ivov 
in the original top view C, then N"N'' perpendicular to o"v" is the final base-line, from which 
are set up first the front view A" by means of C" and A', as in Fig. 83 ; then the view D" 
from the left by means of A" and C", exactly as in the preceding cases. 

139. Fig. 85 is a simple exercise, in which A, B, C are the front, end, and top views of a 
regular hexagonal pyramid, one of whose faces lies in the horizontal plane ; D is a view look- 
ing perpendicularly at the base, which is thus shown in its true form and size, — an expedient 
often of great use ; and C is a perpendicular view of the upper face cod of the pyramid, thus 
showing the true lengths of the slant edges. 

In this figure the circle circumscribing the base of the pyramid is drawn in the direct 
end view D ; and this circle must, evidently, be represented in the foreshortened top view C 
by the curve which passes through the points a, b, c, etc. The base of the pyramid might 
have had twenty sides as well as only six, without introducing any new principle in projecting 
it; and the more numerous the points in the circumference of the circle, the more accurately 
will the foreshortened view of it be determined. But one or two further considerations in 
regard to drawing such projections of the circle are worthy of special note, since they are very 
frequently required in working plans of machinery. 

140. In Fig. 86 the view A shows the edge of a circular disk, situated as was the base of 
the pyramid in Fig. 85 ; and ab in this view also represents the side of the circumscribing 
square, which is drawn in the view D. In the top view C this square will be foreshortened 
into the rectangle shown, and the circle will appear as an ellipse, of which xx and jy/ are the 
axes. Now constructing the view A" , looking horizontally at C in the direction of the arrow 
V, the circumscribing square will appear as a parallelogram abed, to the sides of which xx and 



OBJECTS IN INCLINED POSITIONS. 



73 




74 



MECHANICAL DRAWING. 



yy are parallel : these lines are conjugate diameters, but not the axes, of the inscribed 
ellipse. 

In order to find the axes, we may proceed as follows : Draw a visual ray, that is, a paral- 
lel to the arrow v, through the centre o in the view C. This ray being horizontal, a plane per- 
pendicular to it will be vertical, and will cut the plane of the circle in the line /2^,-which will 
be a diameter, because it passes through o, and will in the view A" appear of its true length. 

Producing hg in the ^'iew C to cut ad and be in vi and ?z, project these points perpen- 
dicularly to ad and be in the view A" , draw mon in that view, and set off oh and og equal to 




:Fig.85, 



the true radius of the circle, given in the view D : then hg is the major axis, since it appears 
of its true length, while every other diameter is foreshortened. 

141. Since the minor axis is perpendicular to the major, draw in the viewy^" a perpen- 
dicular to inn, cutting ab and ed in k and i\ project these points back to the view C, in which 
draw ik cutting the ellipse m. p and q, which latter points being projected again to ik in A" , 
will determine the length of the minor axis required. 

This, however, depends upon the perfection of the ellipse in the view C; and if the great- 
est accuracy is desired, the following method is preferable : Observing that lines parallel to 
XX are not foreshortened at all in the view C, we may transfer the measurements ym, yn 
directly to the view D, and draw mil in that view, also ik perpendicular to it, thus locating/ 
and q in the original position of the circle. These points are then projected to the edge view 
A, thus determining their distances above the horizontal plane AW. Finally, in the view A" 
draw parallels to N"N" , at the distances above it just ascertained in A ; which will cut ik at 
the required extremities of the minor axis. 



OBJECTS IN INCLINED POSITIONS. 



75 




^6 



MECHANICAL DRAWING. 



142. These operations are, evidently, outside of the range of projection pure and simple, 
for they depend upon the mathematical considerations that any projection of a circle is an 
ellipse, and that any two diameters of the circle which are at right angles to each other will 
in the projection appear as conjugate diameters of the ellipse. Had the curves been deter- 
mined simply by projecting a number of points, of course the same result should follow ; but 
the shortest way is always the easiest, and it is perfectly proper to save time by availing our- 
selves of those known properties for finding the axes, and then to construct the ellipse by any 
method that is most convenient. And sometimes it is quite important to draw the curves 
with precision, — as for example in making inclined projections of toothed wheels. 

In many cases, however, no such minute accuracy is necessary, — as for instance in repre- 
senting an inclined cylinder; and in Fig. 87 is shown a shorter process well adapted to such 
purposes. 




Fig. 87. 

143. The circle is shown as circumscribed by two squares, the diagonals of each being 
parallel to the sides of the other : it is clear then that OA and OD, which in the projection 
appear as O' A' and O' D\ will be foreshortened in the same proportion, and so will OB and 
OE. This fact renders it unnecessary to draw the second square in the original or direct view 
of the circle. For, having found the projections of one, Imnr, and of its centre-lines, as in 
Fig. 86, the lengths of the diagonals of the other may be ascertained by a very simple process 
shown in the diagram at the right : draw a horizontal line, and another making an angle of 
45° with it ; from the vertex set off oa, ob, respectively equal to O' A' and O' B' ; at a and b 
erect vertical lines cutting the inclined one in d and e. Set off O' D' = od, and O' E' = oe, and 
draw the sides of the second parallelogram parallel to the diagonals mr, In, as shown. We 
have thus eight points in the required ellipse, and what is of still more service, the tangents 
at those points. 

It may perhaps be added, that short portions only of the lines made use of in this illus- 
tration need be actually drawn in practice ; with due attention to this precaution, this 
method is both neat and expeditious. 

144. Application of the foregoing may be made in constructing the projections of the 
cylinder given in Fig. 88, of which no detailed explanation is needed. 

But in whatever position the cylinder is placed, it is possible to conceive two planes 



OBJECTS IN INCLINED POSITIONS. 



77 




78 MECHANICAL DRAWING. 

tangent to it on opposite sides, and parallel to the direction in which we are looking : these 
planes will touch it along two lines parallel to the axis, and the least distance between them 
will be equal to the diameter of the cylinder. 

From this it follows, that in any projection the apparent diameter will be equal to the 
real diameter; and also that if in any view, as for instance A" in Fig. 88, we first determine 
00, the projection of the axis, then a line perpendicular to that projection at either extremity, 
and equal to the actual diameter of the cylinder, will be the major axis of the ellipse repre- 
senting the corresponding base. 

And it is to be noted also that in Figs. 86 and 88 the ellipses in views C and A" have 
on opposite sides the common tangents ss, tt, perpendicular to N"N" ; which fact, if the first 
ellipse is correctly laid out, is of service in constructing the second, even if the point of 
contact be not determined — which indeed is practically of no consequence whatever. 



79 



CHAPTER V. 
OF THE HELIX, AND ITS APPLICATION IN THE DRAWING OF SCREWS. 



145. The helix is a curve traced upon the surface of a cylinder by a point which moves 
at a uniform rate around the circumference, and at the same time travels uniformly in a 
direction parallel to the axis. But though both motions are uniform, their rates are inde- 
pendent of each other; while going once around the cylinder the point may advance in the 
direction of the axis to any distance, great or small, at pleasure, and this distance between 
the successive coils is called the pitch. The directions of the two motions are also independ- 
ent of each other, and thus the helix may be either right-handed or left-handed : to use a 
homely illustration, there are two directions in which a string may be wound around a stick. 



B 











9 

ft 




5-. 




6 








^ y 




3 






^ 


M 


a 
1 


T^-^C^ 


N 


A 


1^3 




* 5( 


1 




9~" 


"X 


Af 








'^•.^ 


6 


^ \ ■ 




■• y^ 




aS>.__ 


a" 


-^i 






Fig. 



Fig. 90. 



In Fig. 89 are given a side view and a top view of a vertical cylinder. In the top view, 
beginning at A, divide the circumference into equal parts by the points 1, 2, 3, etc., and 
project these points up to the lower base of the side view ; where of course the divisions, 
being unequally foreshortened, will not appear equal. 

In the side view set off along AB equal spaces, numbering them also 1, 2, 3, etc., 
from A. Through each of these points on AB draw a horizontal line, and through the 
divisions on AN draw vertical lines : the intersections of those correspondingly numbered 
will be points upon the helix ^369^. The portion ^436, lying on the front side of the 



8o MECHANICAL DRAWING. 

cylinder, is visible, and shown in a full line ; the portion 69j5 is hidden, lying on the back of 
the cylinder, and therefore shown in a dotted line. 

146. It is easy to see that the visible portion ^43G is divided at the point 3 into two equal 
and similar parts, A'^ being concave toward AN, while 36 is convex toward that line. The 
curvature is most rapid at the points A and 6, diminishing toward the point 3, which is the 
point of contrary flexure. The radius of curvature being infinite at this point, the curvature 
of the helix for some distance on either side of it will be almost imperceptible, and this part 
of the line is practically best drawn with a straight-edge. And the same is true with regard 
to the concealed portion 69j5. 

It is equally apparent that the visible part is exactly similar to the invisible part ; the 
horizontal line %^ is a line of symmetry, and the curve 69i> above it is a reproduction of 63^ 
below it, in a reversed position. 

147. Moreover, the projection of the helix is tangent to the visible contour of the 
cylinder : thus, AB is tangent to the curve at A and B, and NO is tangent to it at the point 
6, which is the vertex of the portion 369, symmetrical about the horizontal 66. 

For this reason it is advisable always to use the bow-pen for drawing a small portion of 
the curve on each side of the vertex, finding by trial and error the proper radius and the 
centre x upon the line of symmetry 66 ; the same radius being of course used wherever 
applicable at other points, as at A and B in this figure. 

Since the curvature changes most rapidly near these vertices, care should be taken to 
find points near each other in these parts, in order to locate the centre x with a reasonable 
degree of precision ; which is absolutely essential to a successful representation of the helix, 
an error at the vertex being more conspicuous than in any other portion of the curve. 

148. Suppose the cylinder to be made of a thin sheet of metal, and cut lengthwise along 
the line AB: it may then be unrolled into a flat sheet, shown in Fig. 90, of which the bieadth 
AB is equal to the length of the cylinder, and the length AA' is equal to the circumference. 
This is called the development of the cylinder; and since the unrolling involves no distortion, 
extension, or compression in any part of the metal sheet, the equidistant points 1, 2, 3, etc., 
on the circumference in Fig. 89 will appear as equidistant points on A A' in Fig. 90; and the 
same is true of the divisions on AB. Therefore, drawing horizontals through the latter and 
verticals through the former, the intersections of the correspondingly numbered ones must be 
points on the development of the helix. From the constant ratio between the bases and the 
altitudes of the triangles thus formed, it follows that this development must be a right lii.?-. 
which in this case is a diagonal AB' of the rectangle, because the pitch of the helix is equal 
to the length of the cylinder. 

149. It is quite plain that this operation might be reversed ; were a sheet of thin paper 
cut into the form of the triangle AA'B', and rolled around a wooden cylinder of the given 
dimensions, its hypothenuse would coincide with the original helix, the vertical lines upon it 
also coinciding with those first drawn upon the cylinder. 

Now suppose the upper edge of the paper cut in the form of the curve AMB' : it is clear 
that this edge when wrapped round the cylinder would noi form a true helix. But we can 
easily determine the projection of the curve which it would form, since the vertical lines on 
the paper would go back to tlieir original places on the cylinder, and their lengths are given 



OF THE HELIX. AND ITS APPLICATION IN THE DP A IVING OF SCPFJVS. Si 

in the development. In constructing the drawing, then, we have onh- to project the points a, 
b, M, c, etc., in which these verticals cut the cui-ve in Fig. 90, horizontally back to the cor- 
respondingly numbered verticals in Fig. 89. 

It follows from the above that the shortest line which can be drawn on the surface of a 
cylinder, between two given points which lie neither upon the same circumference nor upon 
the same right line, is a helical arc. For on the development these points will appear at their 
true distance from each other ; and the right line joining them, being by the conditions 
neither horizontal nor vertical, will become a helix when the developed sheet is re-formed into 
a cylinder. 

150. The helix is sometimes called a linear screw, and may be traced by the point of a 
cutting tool moving endwise at a uniform rate along a cylinder revolving in a lathe, and just 
in contact with its surface. If the tool be fed farther in toward the axis, a helical groove will 
be cut, of a form depending on the shape of the cutting part of the tool ; and if the pitch be 
great enough, a helical projecting ridge will be left, and this is called a screw-thread. We say 
if the pitch be great enough ; for if the longitudinal travel be very slow, it is easy to see that 
this thread will be entirely cut away, and the operation will result in merely turning down the 
original cylinder to a smaller diameter. Bj' var3-ing the shape of the tool, then, screws of 
various forms may be produced, of which the most common are the square-threaded and the 
V-threaded. 

THE SQUARE-THREADED SCREW. 

151. A clear idea of the square-threaded screw may be formed by imagining a square rod 
of lead to be coiled around a cylindrical core, like a string around a stick, leaving a space, as 
wide as the leaden bar itself, between the adjacent coils. Thus in Fig. 91 acif\<s, the section 
of this bar ; after making a half turn it appears in the position cghn, and at the end of a 
whole turn in the position dkon ; since the groove is to be as wide as the thread, fd = a/, and 
ik = ei. Also, af, an, dn,jz will lie upon the surface of a larger cylinder, in which the helical 
groove might be cut as before explained. 

The Hne ae is perpendicular to the axis, which it would cut at v, and if still farther pro- 
longed it would cut the opposite sides of the two cylinders at u and iv. Now ad, or its equal 
fn (being the breadth of the thread plus that of the groove), is the pitch of the screw ; 
in going from a to d, the point a describes a helix abed, lying on the surface of the outer 
cylinder, and in going from c to k, the point c describes a helix ebgk, lying on the surface of 
the smaller cylinder : similarly, / and / describe the helices ///;;/;/, ihlo, and all the other 
curves, for successive threads, are simply repetitions of these. 

152. In order to make the drawing neatly, as few lines as possible should be pen- 
cilled in, and very few are in fact necessary. Having lightly drawn the outlines of the outer 
and inner cylinders, and the axis, an indefinite vertical line through a determines the points e, 
V, u, w ; then, observing that vb — ^ pitch, and zuc = J pitch, the vertices of the helices, as/, d, 
n, I, VI, j, etc., as well as the points h, p, q, etc., on the axis, may at once be set off by means 
of the scale ; and xery short vertical lines only need be drawn w ith the triangles through the 
vertices, on which are located the centres of the small arcs to be drawn with the bows. Let 



82 



MECHANICAL DRAWING. 




o 



now the visible portions of the helices 
for one half of a thread, as sdrp, nq, 
kp, be carefully constructed and 
traced on a piece of tracing-paper, 
on which are also to be drawn the 
axis and the vertical lines dk, no. 
Turn the tracing-paper over and 
trace the lines carefully with a pencil 
of medium hardness, on the opposite 
side, then turn it over again to its 
original position ; by sliding it along 
and adjusting it to the positions of 
the successive threads, the lines on 
its back may be transferred to the 
paper below by going over them 
carefully, with a firm pressure, with a 
pencil. Having thus made impres- 
sions or transfers for the upper halves 
of the threads, turn the tracing-paper 
end for end, and repeat the opera- 
tion for the lower halves. 

153. In the construction of the 
curves particular attention should be 
paid to the fact that the visible por- 
tion ds on the remote side is in the 
projection precisely similar to the 
corresponding portion dr on the 
nearer side, dk being the line of 
symmetry, so that ks ^ kr ; and also 
to the fact that kp is not tangent to 
drpj, but intersects it at p, at an 
angle which may be determined as 
follows : 

The helical arc ab embraces one 
fourth the circumference of the cylin- 
der, and the corresponding linear 
advance is vb, one fourth the pitch. 
On va produced set off vy, equal to 
the quarter circumference, then by is 
tangent to the helix at b ; and the 
angle vyb is the measure of the 
obliquity of the curve. With the 
same pitch, the obliquity will be the 



OF THE HELIX, AND ITS APPLICATION IN THE DRAWING OF SCREWS. 83 

greater the smaller the cylinder : set off then vx equal to the quarter circumference of the 
core, then bx is tangent to the inner helix, and vxb is its obliquity. 

If, in inking in, small portions of the helices near the axes are drawn with the straight- 
edge, these tangents should always be constructed and the triangles set so as to make those 
straight portions parallel to them. 

154. When a very long screw has to be drawn, it will facilitate matters and save time to 
make a template of a thin slip of white holly, cut to the exact form of the helical curve. The 
centre-line should be drawn upon this, and it is used in connection with tlie T-square as shown 
in Fig. 92 ; it is necessary only to slide it along to bring the point o successively to the posi- 
tions b, h,p, etc., in Fig. 91. In making such a template, no attempt should be made to 
carry the helical curve farther than the points of contact with the small circular arcs at the 
vertices, as at f, in, Fig. 91, for neither pencil nor ruling pen can be successfully used to 
draw such sharp curves by any template, concave or convex; beyond those points, the outline 
should be straight, and tangent to the helix. Such small portions of the inner helices are 




'^'^^^^./-■-■^ 



Fig. 92. 

seen in the drawing, that it is seldom or never worth while to make a template for drawing 
them. 

Should proper wood for the purpose not be at hand, it is well to know that such curved 
rulers can be made of heavy card-board or Bristol-board to good advantage : this should 
receive on each side one or two coats of shellac varnish, and when well dried it will then be 
found easy to produce a very satisfactory drawing edge by means of fine files and sand-paper. 

155. The effectiveness of the drawing will be much enhanced by the proper use of 
shadow-lines. The arrow shown in the end view at the left of Fig. 91, being drawn toward 
the centre, is normal to the surface at P, at which point the light is most concentrated ; the 



84 



MECHANICAL DRAWING. 



corresponding point in the side view is therefore the one at which the shadow-line should be 
heaviest : from P to Q'\t may be substantially uniform, but from Q\.o R'\\. should be gradually 
diminished in thickness, merging at the latter point into the unshaded outline. Undeniably, 
a shadow is cast by the helix up to the very highest visible point ; but the surface above P is 
receding, and the shadow-line is therefore tapered off rapidly above that point, in accordance 
with the general principle that the heavier the shadow-line, the more pronounced is the effect 
of relief. 

156. If we suppose an iron screw to be put into a mould as a core, and type-metal to be 
cast around it, it will be seen that this metal will exactly fill the groove. The surface of the 
nut thus formed, then, is identical with the surface of the screw : if now, without unscrewing 
the iron core, we plane off one half of the whole, down to the axis, it is evident that the outline 
of this longitudinal section of the nut will coincide precisely with that of the section of the 
screw. Consequently, if the remaining half of the screw be taken out, the interior of the nut 
will exhibit lines which, from the manner in which it was made, must be identical with those 
portions of the helices lying on the farther side of the screw, and therefore not visible in Fig_ 
91. Such a section of the nut is shown in Fig. 93, and requires no farther explanation, except 




Fig. 93. 



to call attention to the circumstance that all shadow-lines except those lying in the plane of 
the section are omitted, since all the edges which would cast shadows lie upon receding 
surfaces. 

157. The screw shown in Fig. 91 is a single-threaded one; now if we wind the same 
bar around the same core, but double the pitch, the groove will be three times as wide as the 
thread. In the middle of this groove we may wind another bar of the same section, thus 
forming a double-threaded screw. Or, what comes to the same thing, if we cut in the 
lathe a groove of the same size as that shown in Fig. 91, but of twice the pitch, there will be 
left a thread three times as wide, in which another similar groove may be cut. A double- 
threaded screw, with a section of its nut, is shown in Fig. 94. 

Clearly, the groove may be made deeper or shallower at pleasure, and wider or narrower 
than the thread ; also, by trebling the pitch we may add another thread of the same size, and 



OF THE HELIX, AND ITS APPLICATION IN THE DRAWING OF SCREWS. 85 

so on indefinitely. But unless the diameter is increased, the obliquity of the helices will be- 
come greater with each additional thread, until at last it would be impossible to turn the 
screw in the nut if the latter were fixed. 




Fig. 94. 



Examples of such multiple-threaded " quick" screws are found in the so-called "Archi- 
medean " drills, which are rotated by sliding the nut endlong. None of these variations, 
however, involve anything in regard to the making of the 
drawings beyond what has been explained. 

158, The operation of winding a rod or bar around a 
cylindrical core, which we have used as illustrating the formation 
of a screw-thread, is of course actually performed in the making 
of helical springs. The "wire" of which these are made is 
usually either round or square in transverse section, although 
other forms are occasionally employed. 

The appearance of the spring, after withdrawing the core 
around which a square bar has been wound, is shown in Fig. 95 ; 
it requires no special explanation, the visible lines being por-- 
tions of the helices into which the edges of the bar have been 
twisted, and short parts of the rectilinear elements of the outer 
cylinder. 

But if a round bar be thus coiled into a helical spring, its 
axis will be twisted into a helix lying on the surface of a 
cylinder, whose diameter is that of the core plus that of the 
bar. Now suppose a fine wire to be coiled into the form of this 
helix, and a hole of the same s'ize and curvature as this wire to 

pass through the centre of a sphere of the same diameter as the bar. Since the curvature of 
the helix is everywhere the same, the ball could slide along the wire, and in every position 
would just fit a similarly coiled tube of the same diameter. 

Having drawn the central helix, then, describe a number of circles, with centres on this 
helix, and a radius equal to that of the bar ; the envelope, or curve tangent to all these circles 




Fig. 95. 



86 



MECHANICAL DRAWING. 



will be the visible contour of the coiled rod, as shown in Fig. 96. A moment's study will show 
that the line ab of this contour cannot extend beyond the point /, at which it is tangent to 

the outline of the sphere whose centre is c, the vertex of 
the central helix ocp ; and in like manner the line de must 
disappear at /, where it is tangent to the outline of the 
sphere whose centre is ; and similarly in regard to the 
other coils. 

The operation just described is the most expeditious 
~ as well as the most accurate method of determining the 
correct visible contour of the helical coil, and is to be 
employed when the object is of considerable dimensions, 
as in the case of a very large spring, or the worm of a 
still. 







Fig. 97. 
Fig. 96. 
But springs are seldom so large as to render this labor necessary ; under ordinary cir- 
cumstances, the mode of representation shown in Fig. 97 answers all purposes, care beino- 
taken to locate correctly the centres of the small circles, to which straight parallel tangents 
are drawn ; the curvature of the helix being slight, may be neglected. 

THE V-THREADED SCREW. 

159. This may be formed by winding a flexible bar of triangular section, base downward, 
upon a cylindrical core, as shown in Fig. 98. The pitch is here made equal to the base of the 
triangle, so that the sections of adjacent coils, as aef, fgh, just touch each other at the root 
of the groove left between them, where they have the common point/: this forms what is 
called the full sharp single-threaded screw, which of course is actually made in practice by 
cutting the triangular groove with a sharp-pointed tool. 

Now in winding the flexible bar it is at once seen that the outer edge a is twisted into a 
helix alq lying on the outer cylinder, whose outline is avg; and the lower edge e into a helix 
lying on the surface of the core. 

In like manner the intermediate points b, c, d oi the side ae of the section will describe 
helices lying on intermediate cylinders: these will have different obliquities, because they have 
the same pitch, ag or ef. Were these actually engraved upon the surface of the screw, it is 
evident that they would be visible on the nearest side, up to and beyond the points b, c,d, but 
would finally disappear. Since they lie 7ipon the surface, then, and do not go outside of it, 
the visible contour must be bounded by a line ik, tangent to all the helices : the line 
ae, on this side of the thread, is still visible ; but it is not the outline, and therefore is not 
shown in the finished drawing. 



OF THE HELIX, AND ITS APPLICATION LN THE DRA WING OF SCREWS. 87 

160. Similarly, the visible contour of the right-hand side of the thread will be a line 
hn, tangent to all the helices described by the points on the line af. But since it is the 
visible contour, those helices will disappear from view at the points of contact with hn\ the 
outline of the section «/ being thus concealed by an intervening portion of the thread. 

Obviously, the same reasoning and the same process would enable us to determine the 
extreme outline of a thread wliose section was of any other form, symmetrical or otherwise. 

And it is equally clear that when, as is usually the case, the section is symmetrical with 
regard to a line, as aw, perpendicular to the axis, the contour-lines ik and Im will be similar, 
and also symmetrical in respect to the same line. 




In the figure the triangle acf'x?, equilateral, because in the U. S. standard screws the sides 
of the threads and grooves are inclined to each other at an angle of 60°. This being so, it is to 
be observed that the contour-lines are in reality slightly concave; but with the usual propor- 
tions between the size of the thread and the diameter of the core (practically determined by 
considerations of strength), the curvature is too slight to be appreciable. 

161. It suffices, therefore, to draw only the outer and the inner helices, aq and cr, carry- 
ing them well beyond the vertices a and e, and to draw a straight line tangent to these two. 
This tangent should be prolonged to cut the vertical line through a in the point : then hn 
will also pass through o, and the angle worn will be equal to the angle wok\ also, be it noted, 
each of these angles will be greater than wae. 

The contour-lines will be tangent to the outer helix at two points i and /, on opposite 
sides of the vertex and equally distant from it ; and this small arc is best drawn with the bow- 
pen. The thread, then, though actually sharp at the top, does not appear so in a correct side 
view of the screw, when the axis is parallel to the paper, as it usually is. 

In regard to the groove, the case is different; and here it is to be observed that since nj. 



88 



MECHANICAL DRAWING. 



the left-hand side of the next thread, is parallel to ik, it will cut Im at some point, /, on the 
vertical line through/, but some distance above it. 

The groove, therefore, does appear sharp, though not as sharp as it really is ; the bottom 
of it (or the actual root of the thread) is not visible ; and the apparent depth, vp, is less than 
the full depth. In Fig. 98 only the upper half of the screw is shown ; which however is 
sufificient, since if simply inverted it represents the lower half just as well. 

Accordingly, in Fig. 99, which represents the entire screw, 0' k' is parallel to ok, and o'ln' 
parallel to oiii. In order to make the screw enter the nut readily, the thickness of the last 
thread is reduced by turning it off by a plane perpendicular to the axis ; the end of the thread 




Fig. 99. 

is also rounded as shown in the end view, thus producing in the side view two curves tangent to 

each other and to the outer helix at a. 

162. A section of the nut is given in Fig. ico. In relation to this, it is a common error 

to suppose that its contour should correspond with the visible contour of the screw, 

as alpng in Fig. 98. That this is not the 
case will be clearly seen by imagining the 
nut, as before suggested, to be made by 
casting type-metal around the screw, which, 
expanding as it solidifies, must exactly fill 
the groove and fit the screw. A section of 
both screw and nut by a plane through the 
axis will then show but one contour-line 
common to the two ; and this will be com- 
posed of right lines, intersecting each other, 
in the case here illustrated, at angles of 60°, 
with no rounding off or curvature of any 
kind whatever, as shown in the figure. 
Fig. 100. On removing the remaining half of the 

screw the outer and inner helices on the interior of the nut will be exposed ; small portions of the 




OF THE HELIX, AlTl) 




APPLICATION IN THE DRAWING OF SCREWS. 89 



former, near the vertices, will be hidden on account of the twist of the surface, but the latter 
will be visible from vertex to vertex. 




Fig. ioi. 

163. The threads of bolts are seldom made " full sharp," but are usually made flat on 
the top, the groove also being flat on the bottom, as shown in Fig. loi. In the complete 
drawing, therefore, the bottom of the groove is usually seen, as in the figure : the construction 
does not involve any considerations beyond those already presented, but it may be as well to 
call attention to the fact that the contour-line ik in Fig. 98 intersects the upper outline of the 
core at a point s, at a small but yet a perceptible distance to the left of the vertex, e, of the 
inner helix. 

It need hardly be added, that V-threaded as well as square-threaded screws may be made, 
and sometimes are required to be, left-handed as well as right-handed, and with two, three, or 
any other number of threads. Screws are also sometimes made with a different slope on one 
side of the thread from that on the other, or even with threads whose section is bounded by 
curved lines. It is not, however, necessary to illustrate these variations, as due attention to 
the foregoing will enable the student to draw them correctly. 

164. In the preceding illustrations of the application of the helix in the drawing of screws 
and nuts no attention has been paid to the customar)' proportions of diameter to pitch and 
depth of thread. The usual conditions have been purposely exaggerated in order to empha- 
size the peculiarities of the curve. And this has been done with a set purpose : in the 
ordinary course of mechanical drawing for constructive purposes, bolts and nuts are very 
seldom met with which need to be drawn with such minute precision as has been heretofore 
insisted on ; and in Part II a chapter is devoted to the illustration of what may be called 
conventional methods of representing them when used, as they mostly are, merely as a means 
of fastening various parts together. 

But the helix is encountered by the mechanical engineer on much larger scales in the 
construction of screw-propellers ; and in laying out the drawings for these the utmost exact- 
ness in regard to this fundamental curve is necessary. 

No illustration of this application is here introduced, because the drawing of even the 
simplest form of the propeller in use involves problems relating to the intersection of sur- 
faces which could not be reasonably discussed at this stage. In the mean time, a thorough 
understanding of the processes relating to the correct delineation of the helix can be acquired, 
independently of the application of the curve to that or any other purpose. 



9° 



MECHANICAL DRAWING. 



CHAPTER VI. 

INTERSECTIONS AND DEVELOPMENTS OF SURFACES. 

165. In Fig. I02 a vertical hexagonal prism is represented as cut by a plane//. Since 
this plane is perpendicular to the paper, the points in which it cuts the edges 1, 2, etc., are 




seen directly in the front view A, and it is self-evident that the top view C is not affected in 
any way. 



INTERSECTIONS AND DEVELOPMENTS. 91 

The true form of the section will be seen by constructing a view, D, looking perpen- 
dicularly against the plane//, as shown by the arrow. In this view the line 1-4 will appear 
of its true length and parallel to//; and the lines 2-6, 3-5 will be perpendicular to 1-4, and 
also appear of their true length, which is seen in the top view C. Having located the points 
thus indicated, the contour of the section will be an irregular hexagon. 

166. Now suppose the upper part of the prism to be removed, and the lower part to be 
formed of thin sheet-metal. Let this be cut vertically along the edge 1, and unfolded into a 
plane. Since the edges of the lower base lie and remain in the horizontal plane, they will 
extend themselves into one right line, whose length is equal to the perimeter of the base C, 
that is, to six times one of the sides. Thus, in the development shown at the left, the edges 
will be represented by equidistant vertical lines, and the lengths of these lines will be the 
same as in the front view A of the truncated prism. The upper extremities of these verticals 
are joined by right lines, which completes the development of the vertical surface of the 
prism. If desired, D' , a copy of D, and C , a copy of C, may be attached as shown : the 
entire outline thus formed being cut out of card-board, and the lines of junction being cut 
half through the thickness with a sharp knife, the whole may be folded up into a very satis- 
factory model. 

167. Let it be required to find the shortest path on the vertical surface of the prism, 
from the point a to the point d. In the development these points fall at a' and d' ; the least 
distance between them is the right line ad', which cuts the edges 2' and 3' at V and c' ; pro- 
ject these back to the original positions of those edges in view A, and the broken line abed 
represents the path required. 

This problem of " the shortest path" may be similarly treated in the case of any solid 
whose surface is capable of development into a plane, provided that in the development a 
straight Hne can be drawn between the two points. 

168. If we imagine the prism in Fig. 102 to be inscribed in a circular cylinder, the circle 
which circumscribes the hexagon in the top view C will be the base of that cylinder, the 
edges of the prism will be vertical elements, and the true form of the section will be the 
ellipse shown in a dotted line in view D ; also, the development of this section will be a curve 
passing through the points 1', 2', 3', etc., of the development of the prism. 

In order to define these curves more accurately, it is necessary only to divide the circum- 
ference in the top view into a greater number of parts, and by thus increasing the number of 
elements, as shown in Fig. 103, to locate more points in the curves. The details of the 
process have already been explained in connection with Fig. 90, and need not be repeated. 
Also, it was shown (149) that the " shortest path" between two points on the surface of the 
cylinder is in general a helix. That is to say, a helix drawn upon a cylinder develops into a 
right line when the cylinder is unrolled into a plane. 

169. Conversely, if an oblique right line is projected upon a cylinder, its development on 
unrolling the cylinder will be the projection of a helix. Thus in Fig. 103, ab represents such 
a line projected upon the cylinder in the front view A ; draw in the development the hori- 
zontal line da" , and through b' a line // parallel and equal to it : then the curve a'b'a" is the . 
projection of a helix whose pitch is a' a", lying upon a cylinder whose diameter is «'/, the axis 



92 



MECHANICAL DRAWING. 



being the horizontal xx passing through 3 in view A. Moreover, the tangent tt at 3" is 
parallel to ab or //. The precautions and expedients mentioned in relation to the drawing 
of the helix therefore apply as well to the drawing of this development of the section of a 
cylinder by a plane. 

Since the ellipse is perfectly symmetrical about both axes, it may be turned end for end ; 
and the two parts of a cylinder thus cut by a plane making an angle of 45° with the axis 
may be joined together as at V, making what is known as a "square elbow." By using other 




Fig. 103. 



angles, the pieces may be put together at different inclinations, as shown at W, which repre- 
sents a " three-section elbow." To lay out the sheet for the middle piece, cut it transversely 
by a plane rr : the circle will develop into a right line, and the ordinates are measured each 
way from this in order to determine the contour. 

170. In Fig. 104 the hexagonal pyramid is cut by the plane />p. The top view of the 
section is an irregular hexagon, determined by projecting the points 1, 2, 3, etc., of the front 
view A vertically downward to the corresponding edges in view C ; after which the view Z> 
is constructed exactly as in the case of the prism. Fig. 102. 

In the development the vertex of course remains a point; and since the edges of the 
pyramid are of equal length, their lower extremities will lie in the circumference of a circle 
whose centre is ■?'', and radius 7''u', equal to v/i : the chords h'j/i', vi'r\ and so on, are equal to 
each other and of the true length, niii, vir, of the edges of the base. 



INTERSECTIONS AND DEVELOPMENTS. 



93 



The true distances of the points 1 and 4 from the vertex are seen in view A ; therefore 
in the development v'V = v\, and v'^ = vL But while vm is actually equal to vn, it is 
foreshortened in view A. All parts of it are, however, foreshortened in the same proportion ; 
therefore drawing through 2 a liorizontal line cutting fn, we ha\e <:'2 " as the true distance 
from 2 to the vertex, and the true length of t'3 is found in like manner. Therefore set off 
7''2' = z'2", v'3' = t'3", and the broken line 1'2'3'4:' is the development of the section of the 
pyramid. 

The representation of the shortest path from a to ^ is found by reversing this process : 
the right line a'd' cuts v'm' and v'r' in ^' and c' ; set off z'd", vc", respectively equal to v'i', 







v'c', on vs, and draw horizontals through d" and c" to locate l> and c: then the broken line 
adcd is the path required, which will also be visible in the top view. 

171. If a cone be circumscribed about the pyramid, its base will be the dotted circle in 
the top view of Fig. 104, and the edges of the pyramid being rectilinear elements of the 
cone, the true form of the section will be the dotted ellipse in the view D : this section will 
also be projected as an ellipse in the view C, and in the development will form a curve 
passing through the points 1', 2', etc. 



94 



MECHANICAL DRAWING. 



The construction in the case of the cone is shown in Fig. 105 : if we draw any rectilinear 
elements, as viii, vr, they will be cut by the plane at points 1, 2, seen directly in the front 
view, whence they may be projected to the top view. But in this top view these projecting 
lines will cut the elements very acutely in the neighborhood of vs, so that the curve cannot 
thus be very accurately defined. In this region another process is preferable: suppose the 
cone cut across transversely by a plane If; the section will be a circle, which may be drawn 
in the top view. Since the planes //, pp are both perpendicular to the paper in the front 
view, the point q represents their intersection, which in the top view will be seen in its true 




Ti^.105. 



^ 


f 




/ 


V 


\/X) 


/ J 


%/V 




17- -- 


m^ 




1 r^ 


^v 


M W-h 




/ 


' \ \\ 


% 






v/f 




i\ w 


vv-i 

1 
1 


s 




/ 


; 


c 


~"^ 


at 


m 


1 


1 


^i 1 


V 


1 


i 


c 


^^ 1 t) 



length as a chord qq of the circle just drawn. This operation maybe repeated as many times 
as is necessary: and it is needless to say that any such chord will also be seen in its true 
length in the view D. 

172. Since the section of the cone by the plane is known to be an ellipse (when, as in this 
case, all the elements are cut), it is perfectly legitimate to determine the axes, and construct 
the curve by any convenient method. It is clear that ab in the front view is the length of 
the major axis in the view D ; to determine the minor axis, bisect ab, and pass a horizontal 
plane gh through the point of bisection, by means of which find the chord cc in the top 
view as just explained : this will be the minor axis of the ellipse seen in the top view, as well 
as of the true elliptical section D. 



INTERSECTIONS AND DEVEIOPMENTS. 95 

It is clear that, as in the case of the cylinder, the ellipse may be turned end for end, and 
the upper part of the cone joined to the lower, as in the small diagram U \ the vertex going 
from its original position v to the new one v' , by revolving around an axis ini, perpendicular 
to the plane of the ellipse at its centre. 

173. The developed surface of the entire cone will be limited by two rectilinear elements, 
including a circular arc whose radius is the slant height, and of a length equal to the circum- 
ference of the base. Having described with that radius an indefinite arc about v' , lay off 
upon it, by the process of Fig. 34, an arc equal in length to a convenient fraction of the cir- 
cumference of the base, rectified as in Fig. 33, and step this off the required number of times, 
to determine w'ti'zv" : this done, points in the curve a'b'a" are located precisely as in the case 
of the pyramid. Fig. 104. In relation to this, it is to be particularly noted that as the 
tangents to the ellipse at a and b are perpendicular to the elements through those points, so 
the tangents to the developed curve are perpendicular to v'a' , v'a", v'b'. 

The "shortest path" from ^ to ^ is determined exactly as in the case of the pyramid. 
Since in the development d'k' cuts both v'w' and v'n' obliquely, it is to be observed that in 
the front view the projection of this path will be tangent to vzv at d, and to vn at k. The 
highest point of this projection may be found as follows: The least distance of k'd' from the 
vertex is the perpendicular v'o' let fall from v' in the development ; set off vo" = v'o' , on vn 
in the front view, and a horizontal through 0" will be tangent to the path at its highest point. 
Produce v'o' to t' , rectify the arc 7t't' , and set off an arc nt of the same length in the top 
view; project t to the line of the base in the front view: then tv will be the element corre- 
sponding to t'v' in the development, and will cut the horizontal through 0" in 0, the required 
point of tangency. 

174. These problems of development are of direct practical use to the workers in sheet- 
metal. Thus if the cone be cut off by another horizontal plane xy, the frustum zvxyn forms 
the body of a funnel ; if in Fig. 103 the plane// cuts the elements of the cylinder at an angle 
of 45°, the two portions of the solid may be joined to form the "square elbow" V ; and the 
development is the shape of the metal sheet from which each piece is made. By using other 
angles, the cylindrical pieces may be joined as at W in Fig. 103 ; in this case an imaginary 
right section at rr will develop into a right line, and by measuring each way from this the 
contour is determined. 

Theoretically, it makes no difference along what element the cylinder or cone is cut ; 
but practically it should, when there is no reason to the contrary, be cut so as to form the 
shortest seam. 

Again, the drawing of a bevel-wheel involves the development of a cone, upon which 
the outlines of teeth are laid out ; the correct representation of the teeth of the wheel 
depending upon the forms which these outlines assume when the developed sheet is re-formed 
into its original cone. 

176. It will be understood from the preceding examples that in the development of a 
surface by unfolding or unrolling it into a plane there is no extension or compression, or dis- 
tortion of any kind. Lines of course change their forms, but not their lengths: if they inter- 
sect on the original surface, they will after development intersect at the same angle ; if, on 



96 



MECHANICAL DRAWING. 



the other hand, they are originally tangent, they will remain tangent. The surfaces of solids 
bounded by planes are developed by unfolding, the various faces turning upon the edges as 
upon hinges ; the surfaces of cylinders and cones, as we have seen, are developed by unrolling, 
the only difference being that the number of edges is infinite. These are the only curved 
surfaces commonly met with that are capable of development ; but there is a third kind, of 
which a single example will be sufficient. 

176. It is a familiar fact that if a piece of paper in the form of a right-angled triangle, as 
AA'B', Fig. 90, be wound around a cylinder so that the base of the triangle forms the cir- 
cumference of the base of the cylinder, the hypothenuse will form a helix: thus AB' of Fig. 
90 becomes the helix ^4369^5 of Fig. 89. In Fig. 106 the paper is shown as partly unwound ; 
i^m being equal to the quadrant a'c' of the base, it is ap- 
parent that the hypothenuse dc is tangent to the helix 
at c, and equal in length to the helical arc ac. It is clear 
that as the unwinding progresses the point i will trace in 
the plane of the base the involute a'd'r' of the circumfer- 
ence ; the upper edge of the paper will in every position be 
tangent to the helix, and these different tangents must lie 
upon a continuous surface, of which a limited portion is 
represented in Fig. 107. It will be recognized that this 
surface somewhat resembles that of a cone ; but as the ele- 
ments, instead of meeting in one point, are tangent at 
various points to the helix, the surface goes on, expanding 
as it rises, and winding about the cylinder in convolutions 
like those of a sea-shell. 

177. The helix itself.winding equably around the cylinder, 
has everywhere the same curvature. In Fig. 106 3c cuts 
ag; the projected outline of the cylinder, at ; draw at this 
point a perpendicular to dc, cutting at ^ a horizontal line 



through c : then ce is the radius of curvature of the helix 
acd. 




Fig. 106. 



Ascertain in this way the radius of curvature, cj, of the helix in Fig. 107 ; and about any 
point O, Fig. 108, describe a circle with this radius. If then the helix be supposed to be made 
even of wire having a sensible thickness, a portion of it equal in length to the circumference 
of this circle can "have the twist taken out of it," and be made into a ring which will lie flat 
on the paper ; and all the tangents to the helix will also lie in the same plane, and be tangent 
to the circle. Since the length of each tangent, in the formation of the surface, was equal to 
that of the helical arc from which it was unwound, and remains the same after the develop- 
ment, the involute a'd'f in Fig. 107 will become the involute of the new circle in Fig. 108. 
Making the tangent AH in this figure equal in length to the circumference, then, the piece 
AHGFBA may be cut out of paper or thin metal, and (the circle being also ciit out) it can 
be bent up, the circumference fitting the helix acde in Fig. 107, the curve ABFG at the same 
time contracting into the involute a'b'f'g', and thus the plane will assume the form of the 



INTERSECTIONS AND DEVELOPMENTS. 



97 




98 MECHANICAL DRAWING. 

original surface. Produce be in Fig. 107, set off bk = AH, and draw a horizontal through k, 
cutting the helix in Ji, which will be the limit to the height of the part of the surface of which 
Fig. 108 is the development. No more than this can be cut out of a single sheet ; but since 
the edge HA is lifted out of the plane in forming up the surface, another piece, shown on a 
smaller scale in Fig. 109, can be joined to the first one along that edge, and so on inde- 
finitely. 

178. Let it be required to find the shortest path upon the surface from x to w, in Fig. 
107. Since x lies upon be, its position X \x\ Fig. 108 will be upon the corresponding tangent ; 
make ^C" equal to the helical arc ac: then the tangent CB is the one required. Similarly, 
make ACDE equal to the helical arc acde : then the tangent EG is the position of eg in the 
development, and upon it IF is located. 

We now find that a straight line cannot be drawn upon this development from X to 
W, because such a line would cross the circle, within which the surface has no existence, 
thus presenting a singular exception to the general rule. Draw PFF tangent to the circum- 
ference: then the shortest path is composed of the two tangents XC, VW, and the circular 
arc CF between the two point3 of tangency. 

On the original surface, then, this path is made up of the portion xe of the element pass- 
ing through x\ of the portion ev of the helix; and of a curve vw tangent to the helix, — which 
can be constructed by drawing in Fig. 108 tangents to the arc VE, as DF for instance : these 
are the developed positions of elements, and cut VW in points which, when they are restored 
to their original locations on the surface, lie upon the required line. 

179. From the preceding examples it will have been gathered that in order to construct 
the intersection of a surface with a plane it is necessary to find the points in which the plane 
cuts a series of lines drawn upon the surface : and it is clear that these lines should be the 
simplest that the nature of the case will admit of, unless the intersections are too acute to 
give reliable determinations ; which may happen, as we have seen in the case of the cone, when 
the rectilinear elements alone are depended on. Also, it will be noted that the object and the 
plane are represented in such a position that in one of the views the plane is seen edgewise : this 
greatly facilitates the construction, and exhibits most clearly the reason for each step. If 
after the section is made a view of the object from any other direction or in any other 
position is needed for any reason, it can readily be made by the processes explained in 
Chap. IV. 

180. Fig. no represents a solid of revolution ; that is to say, one that can be turned in 
the lathe ; every transverse section is a circle, which in general is the simplest line that can be 
drawn on such a solid. 

In determining the intersection of this object with the plane // these circles are made use 
of precisely as in the case of the cone (171). Thus the point a in the front view represents a 
chord in the circle whose radius is ik, and this chord is seen in its true length in the top view 
and also in the view D, which shows the exact outline of the section. So, again, c and d in 
the front view represent the chords ce, dd, in the circle whose radius is hn : e in the front view 
coincides with d, but represents a chord ee of the circle whose radius is In, n being the point 
at which the curved outline of the base of the object is tangent to the prolongation of the 



INTERSECTIONS AND DEVELOPMENTS. 



99 



horizontal line Im. Thus, since the upper side of the base has a narrow plane ring whose 
breadth is mn, the small portions de, de in the section are straight lines. 

It is hardly necessary to point out that in actually constructing the section very small 
arcs only of the circles used need be drawn, and the chords should not be drawn at all, the 
extremities alone being marked. 




181. Fig. Ill represents also a solid of revolution, whose outline ^/ is an hyperbola 
whose conjugate axis // is the axis of the solid. It may also be formed by the revolution 
about the same axis of the two lines aob, cod, which in the front view coincide with the 
asymptotes: these are in this view parallel to the paper, and in the top view coincide in one 
line a'b' , the distance I'o' being equal to io, the semi-major axis ; in the side view B they also 
coincide in one line be, and obviously they determine a plane parallel to //, which, while cut- 
ting these two lines out of the surface, is also tangent to it at the point o. 



loo MECHANICAL DRAWING. 

It is apparent that the point b describes the circle of the lower base, whose radius is l'b'\ 
also that o describes the circle of the gorge, ii, its radius I'o' being also taken directly from the 
top view. Every intermediate point on ab follows the same law, so that if this line and the 
axis are given or assumed, the hyperbohc outline is most expeditiously drawn by the method 
of Fig. 48. 

The section of this surface by an inclined plane pp which cuts all the elements is an 
ellipse, of which the major axis is xy. Bisect xy at r ; a transverse section through this point 




is a circle with radius st, in which r represents the horizontal chord r'r' in the top view, and 
this is the minor axis ; from which data the view D is constructed directly. 

182. Drawing a new base-line N" N" , let it be required to make a view C" , looking per- 
pendicularly down upon the new horizontal plane thus indicated, as shown by the arrow. 
The axis will appear as a line parallel to N"N", and the upper base as an ellipse of which the 
minor axis is h"h", the foreshortened projection of hh, the major axis m"m" being seen in its 
true length, that is, equal to hh. The lower base will appear as a similar and equal ellipse, 
and the circle of the gorge as a smaller one, constructed in like manner. So also any inter- 



INTERSECTIONS AND DEVELOPMENTS. 



101 



mediate transverse sections will be projected as ellipses, small portions of which are shown at 
ti, V, and the outline will be limited by a curve tangent to all these ellipses. 

This illustrates very clearly an important principle of universal application, viz., that the 
visible contour of any object, in any projection whatever, is the envelope of all the lines 
which can be drawn upon its surface. 

183. Fig. 112 represents the "stub end " of a connecting-rod ; it is rectangular in section, 
as shown hy f'kmn in the end view, and is joined to the cylindrical neck of the rod by the 
surface of revolution, whose outline WY is usually, as here shown, a circular arc. We have, 
then, to determine the intersections of this surface by two planes//, rr, parallel to the axis. 

A transverse section at « is a circle, which in the end view is seen to be cut by the plane 




Fig-.ia. 



pp at b' , which is projected back to b, a point in the required curve ; in like manner the point 
d is located by means of a transverse section at c, and so on. 

The plane rr cuts the outline at x, which determines the vertex x" of the curve seen in 
the top view. This plane intersects pp in the line forming the upper front edge of the stub 
end, which appears in the end view as the point /"'. A circle described about ^through f 
cuts the vertical centre-line in /, which, projected back to the contour at e, gives the location 
of a transverse section from which, by the preceding process, would be determined the point 
/in the front view, corresponding to f" in the top view. A transverse section at any point as 



I02 



MECHANICAL DRAWING. 



o, between x and e, is a circle, in which Ji'h' in the end view is a chord, seen in its true length 
as h"h", in the top view, perpendicularly above o. 

184. Fig. 1 12 represents the case, very common in practice, in which the thickness of the 
stub end is equal to the diameter of the neck of the rod, so that the plane// is tangent to the 
neck, as seen in the end view, and also to the concavo-convex surface of revolution at the 
point V. In this case the two symmetrical curves vbdf, vs intersect acutely at v ; the angle 
of intersection may be determined as follows : Let Q be the centre of the contour curve WV; 
produce QW to L, bisect QL at /, and upon it as diameter describe a semicircle cutting the 
axis in G. On vQ set ofif vH ^ vG; about v describe an arc with radius vW, and draw HJ 
tangent to it : then vJT, perpendicular to HJ, is tangent to vs at v. 

If the stub end is thicker than the neck of the rod, the intersection by the side plane will 
be a continuous curve, as fvs in Fig. 113, which will have a vertical tangent at v, and will 



: 


-1 


-- 


-"^^/^ 




) 






] 




V 


^^~^\ \s 




V 


ST ~ 





Fig. 113. 



usually have a point of contrary flexure, as here shown ; the case is analogous to that pre- 
sented by the plane rr in Fig. 112, and requires no farther explanation. If the section of the 
stub end be square, as it sometimes is, the curves on all four sides will of course be exactly 
alike. 

185. Approximate Developments. — It has already been pointed out that only surfaces 
of single curvature, such as the cone and cylinder, are capable of development : a warped or 
twisted surface like that of a screw, or a double-curved one like that of an e^g, cannot be 
transformed into a plane without distortion. Still, approximations may be made, and are 
often of use, as in the laying out of maps, and in determining the forms of boards for covering 
hemispherical domes. 

This last may be accomplished in two ways illustrated in Fig. 1 14. In the first method 
the boards are laid horizontally, forming zones, represented in side elevation and in plan in 
quadrants I. and II., respectively. Let AB, in the enlarged diagram on the left, be half tht'» 
arc measuring one of these zones, and AC the thickness of the board. When in place, it is, 
evident that the edge DB should coincide with a radius DL of the hemisphere, just as CA 
coincides with a radius CG. At C and A draw perpendiculars to CG, and on them set off CF, 
CE, respectively equal to the arcs CD, AB, and draw the radial line FEH: then AEFC is 
one half of the transverse section of the board. 

Let a in quadrant I. correspond to A in the large diagram : at this point draw a tangent 
to the outline of the hemisphere, and produce it to cut the axis '\x\ 0; on ao set off ab' , ae' , 
respectively equal to AB and AE^ and through the points b' , a, e' describe circular arcs with 



INTERSECTIONS AND DEVELOPMENTS. 



103 



centre 0. In the illustration it is assumed that the board is to cover one quarter of the zone : 
therefore the arc al is made equal to the quadrant al, b'k' equal to the quadrant bk, and the 
length of e'i' equal to that of the quadrant ci. This gives the approximate development for 
the inner surface of the board : the breadth of the outer surface, indicated by the exterior 
lines, is of course equal to twice CF. 

186. In the second method the hemisphere is covered with boards cut in the form of 
gores, the edges, when in place, forming meridians extending from the equator to the pole, 
as shown in plan and elevation in quadrants III. and IV. 

To lay out a gore, erect the vertical r'p', equal in length to the quadrant rp, and divide 




ri^.ii4. 



it into equal parts, corresponding to the divisions of rp by the points 1, 2, 3, etc. In the 
plan these latter points are for convenience represented upon the bisector /i-, and through 
them are drawn circular arcs terminating in the edges of the gore. Then in the development 
make the horizontal x'y' , at 2', equal in length to the arc x^y in the plan ; make ?/'4V equal 
to the arc 2i-^v, and so on : the extremities of these horizontals are points in the contour 
required. 

It need hardly be stated that in the application of either method the zones or the gores, 
as the case may be, should be narrow, and the boards thin ; the breadths have been purposely 
made much greater than they could be in practice, in order to exhibit the processes 
clearly : their numbers should be at least four times as great, to give satisfactory results. 



INTERSECTIONS OF SURFACES. 



187. In Fig. 115 is shown a vertical cylinder, penetrated on the left by an inclined 
one, the two axes meeting, and on the right by a horizontal one, whose axis is not in the 
same plane with the other two, but a little in front of that plane. 



I04 



MECHANICAL DRAWING. 




INTERSECTIONS AND DEVEIOPM ENTS. 105 

Beginning with the latter, we observe that the point a in view B represents the line ab in 
views A and C. And in the top view we perceive that this line pierces the vertical cylinder 
at b, which projected to the front view determines one point in the curve of intersection. 
Similarly, c and e in view B represent the front and rear elements, and in view C these ele- 
ments pierce the cylinder at d and f, which projected to view A determine the vertices, at 
which the curve has vertical tangents. If in the top view we draw any element as lo, it will 
be represented in view B by a point /, whose distance It from the vertical plane through a 
is equal to la in view C: this determines the height of lo in view A, to which line the point 
is projected from view C, as before, and so on for as many points as may be required. 

Clearly, (5 is the highest point, at which the curve is tangent to ad, hut it is not the 
extreme right-hand point, which is found thus : the right-hand element of the vertical 
cylinder is in view B coincident with the centre-line, which pierces the horizontal cylinder 
in li, which projected to view A locates the point in question. 

188. In the case of the inclined cylinder, the highest and lowest element are at once 
seen to intersect the left-hand element of the vertical cylinder at i and J; the nearest 
element is seen in view C to pierce it at n, which is projected to view A, as before. This 
inclined cylinder is circular, as shown by the direct end view D ; being cut off at the upper 
end by a horizontal plane, the section appears as an ellipse in view C. But this does not 
affect the mode of operation : if we draw upon it in the top view any element as hv, the 
point li is projected to Ig in view A, a parallel to the axis is drawn through that projection, 
to which the point v is projected from C, thus locating a point in the curve. 

Producing 71U in the top view to cut the ellipse in r, it will be seen that there is another 
element directly beneath vh, on the lower part of the cylinder, from which the point q is 
determined in the same way. 

189. The development of the inclined cylinder is also shown in Fig. 115. A transverse 
section at// will be a circle, and will develope into a right line, from which the distances to 
the points i^j.g, k, r, q, etc., will be the same as their distances from pp in the front view. 
In the development // is one half the circumference in view D\ and to save labor it is 
advisable to subdivide the semi-circumference into equal parts, and draw in A elements 
corresponding to these points of subdivision; then // in the development is to be divided 
in the same manner : thus in the illustration the semicircle img, view D, and its rectification 
// in the development are each divided into four equal parts. This cylinder being cut along 
the upper element ^/l', not only is the seam the shortest possible, but the development is 
bounded by curves principally convex; which is a great practical advantage in cutting out 
a metal sheet: it is also symmetrical with respect to the vertical if, which greatly facilitates 
the drawing of the outline. 

190. In the development of the horizontal cylinder, shown in the upper part of Fig. 1 16, 
symmetry is secured by cutting the pipe along the rear element <r/in view C of the preceding 
figure : it is not the very shortest, but the difference is not great enough to cause any 
practical dif^cultyc In this development cc, of course, is equal in length to the semi-cir- 
cumference cae in view B of Fig. 115; a is the middle point of ce; and cd, ab, zvh, ef, 
the ordinates of the curve, are equal to the corresponding elements in view C, the distance 
aiv being equal to the arc ah in view B. 



io6 



MECHANICAL DRAWING. 



These ordinates, it is to be understood, are given only for illustration ; in practice the 
proper course is to divide the semicircle cae, and its rectification ce in the development, into 
the same number of equal parts, draw corresponding elements in view C, and set up the 
corresponding ordinates of the lengths thus determined, as above advised in reference to 
the inclined cylinder. 

191. In developing the vertical cylinder of Fig. 115, the "shortest seam" would be made 
by cutting it down along either the right-hand or the left-hand element ; but this would lead 
to the very awkward result that a part of one or the other of the openings would be in one 




Fig. 116. 

end of the sheet, and the remaining part in the other. In the lower part of Fig. 116, then, 
the development is made on the supposition that the cut is made down the remote element, 
y, in view C" of Fig. 115, and the cylinder unrolled to the right and left, so that the nearer 
element xx appears as the vertical centre-line. The opening for the horizontal cylinder will 
then appear on the right, its vertex d being at a distance from xx equal to the arc xd, and 
its length being equal to the arc dbf in view C. It will be symmetrical about a horizontal 
line df, at a height above the base equal to that of the axis of the cylinder. The ordinates 



INTERSECTIONS AND DEVELOPMENTS. 



10/ 



of this curve are equal to those of the circle in view B : thus, making db = arc dob in view C, 
the greatest ordinate bb is equal to the ordinate at a, that is, to the radius ; making bh = arc 
bh in C, set up ]iJi ^ ordinate at //, in view i>, and so on. In making the construction the 
systematic course previously suggested should be adopted, (^ in the development and the 
arc dbf, being divided into the same number of equal parts, and corresponding ordinates in 
view B being located, just as the one at / was from the point o in view C. 

The opening for the inclined cylinder will be on the left, and symmetrical about a 




Fig. 117. 



Fig. 118. 



vertical line midway between xx and the end of the sheet. Its greatest breath iin will be 
equal to the rectified arc nkn in view C, and the distances of all points in the curve above the 
base will be the same as in view A. 

192. If the horizontal axis in Fig. 115 is moved back, nearer to the vertical one, the 
points </and/in view A will approach each other, d moving to the right and /"to the left, 
until, when the axes meet, the curves db and/// will coincide, as shown in Fig. 117; Avhich is 
a case very often occurring in practice, a familiar example being that of a boiler with a steam- 
drum. If at the same time the two cylinders are of equal diameters, the intersection will 
assume the form shown in Fig. 118, being composed of two plane curves db, db' , each plane 
making an angle of 45° with either axis; which indeed might have been inferred from consid- 
eration of Fig. 103. 

If, on the other hand, the horizontal axis be brought forward, the point d \n Fig. 115 will 
go to the left, reaching the vertical centre-line when the element cd oi the horizontal cylinder 
intersects the element xx of the upright one. If the axes be still farther separated, there can 
no longer be a complete penetration, and the horizontal cylinder must be carried past the 
other, the effect being shown in Fig. 119. None of these variations involve any new prin- 
ciple, either in determining the intersections themselves or in their development. 

193. In Fig. 120 is shown a cone penetrated on the right by a horizontal cylinder, the 
axes meeting, and on the left by a vertical one, the axes being parallel. In regard to the 
former, the highest and lowest points, /and ^, of the intersection are seen directly in the 
front view. Draw in the three views any element vc of the cone ; in view B this is seen to 
pierce the cylinder at a and b, which points are projected to view A, and thence to view C. 
Draw an element vq tangent to the circumference of the cyHnder in view B ; the point of 



io8 



MECHANICAL DRAWING. 



contact / is an important limit to the curve in the other views, in which also it will be a point 
of tangency. 

Otherwise : Any horizontal plane, as for instance the one through d in view B, will cut an 
element from the cylinder and a circle from the cone. The circle in this instance has the 
radius //,;, and the element is de ; in view C these are seen to intersect in e, which is projected 
to view A. 

So, too, in the case of the vertical cylinder: the horizontal plane through o, for example, 
cuts from the cone a circle of known radius, which in the top view is seen to pierce the 
cylinder at /, which is projected to the horizontal through o in view A. Or had an element 
vr been drawn in view C, piercing the cylinder in i, the same point would have been located 




Fig. 1 19. 

by projecting i from C to vr in view A: as it should by this time have been perceived that if 
a point lies upon any line, its projections must lie upon the corresponding projections of the 
line. Here too a limit is found by drawing an element vp of the cone tangent to the circum- 
ference of the cylinder in view C : the point of contact s is then projected to vp in view A. 

194. Let the cone be cut along the remote element, vx' in view B: then in the develop- 
ment, shown at the right, the arcs xx' , xx" are each equal to half the circumference of the 
base, and are bisected by vin, vn, about which the openings are symmetrical. The arcs inrp, 
ncq are equal to the corresponding arcs in view C. The vertices/",^ are set off by direct 
measurement of z<f, vg, in view.^ ; the points a, b, t are placed at their true distances from 
the vertex, va' , vV , vt' in view B. The developed opening for the vertical cylinder is laid 
out in a similar manner; in regard to this it may be noted that a check upon the accuracy of 
the work is found in this, that the length of the arc ii' should be the same in view C and in 
the development. A system like that previously recommended is applicable to this and 
all similar cases, — the arcs Dip, nq being divided into equal parts on tiie circle of the base 
and also in the development, and intermediate elements drawn through the points of division. 



INTERSECTIONS AND DEVELOPMENTS. 



loo 




IIQ 



MECHANICAL DRAWING. 



The developments of the two cylinders are also shown in Fig. 120; they will be at once 
recognized, and the construction needs no explanation. 

195. In Fig. 121 is represented a vertical cylinder intersected by a cone whose axis is 
inclined and cuts that of the cylinder; the points/, ^ of the intersection are at once seen in 
view A. 

Extend the cone to any convenient distance within the cylinder, and limit it by a trans- 




verse plane mn : the base thus formed is a circle, seen in its true form and size in view D, and 
it is projected as an ellipse in the top view C. In this view C, the visible contour of the cone 
is limited by the element vi tangent to the ellipse, which pierces the cylinder at a. Find the 
point of tangency i, project it to i on mn in view A, in which view draw vt and project a 
upon it from view C: this gives the left-hand limit of the curve. 

Draw any element as vd in views D and A, from the latter project d to the ellipse in view 
C; in this view the element thus determined is seen to pierce the cylinder in c, which is pro- 
jected back to 7'5 in view A. In this way any number of points may be determined ; as a 



INTERSECTIONS AND DEVELOPMENTS. 



II I 



check upon the accuracy, it should be noted that the ordinate br is of the same length in 
views C and D. 

In view C, vb cuts the ellipse also in //, which being projected to inn in A, shows at once 
that there is another element vh on the lower part of the cone, vertically under vb, and pierc- 
ing the cone at a point c vertically under c. Or it may be inferred, since the axis of the 
cylinder is vertical, that a plane through the vertex of the cone, parallel to that axis, cuts 
two elements from the cone and one from the cylinder ; and these lines intersect in points of 
the required curve. 

The process would evidently be the same, were the axes in different planes ; common- 
sense indicating that both should be placed parallel to the paper in the front view. And the 




development of these surfaces should now present no difficulty to the student, for whose 
ingenuity the operation is accordingly left as an exercise. 

196. Fig. 122 shows a hemisphere penetrated hy a vertical cj'linder. Any plane parallel 
to the paper in the front view will cut a circle from the sphere, which in that view will be seen 
in its true form and size : it will cut the cj^linder in two vertical elements if it cuts it at all, 
and these will intersect the circle in points common to both surfaces. Thus the plane seen 
edgewise as mn in the top view, cuts from the c\-linder the extreme right- and left-hand 
elements, which intersect the circle cut from the sphere, seen as rr in the front view, in the 
points o and/ of the curve of intersection. 

Any horizontal plane, as gs in the front view, cuts the sphere in a circle, which in the top 



112 



MECHANICAL DRAWING. 



un. 



view is seen to pierce the cylinder in h and c, which, projected back to gs in the front view, 
locate two other points in the curve. If in the top view we draw a circle as ^tangent to 
the circumference of the cylinder, we can ascertain, by projecting /"to the outline of the 
sphere in the front view, the level of the plane by which that circle was cut out, and thus 
locate d, the lowest point of the curve: in like manner the highest point b is determined. 

197. In Fig. 123 let the two lines macii, oacp revolve together about the axis op. These 

lines are tangent to each other at a, and cut each other at c, 
and will do so throughout the revolution, so that the two 
circles ab, cd are common to the two surfaces thus generated. 
Which illustrates a general truth, viz., that if two surfaces 
of revolution, having a common axis, touch or cut each other 
at any point, they will do so all round the circle described by 
that point. In the illustration both the outlines are curved, 
but clearly either or both might as well have been straight. 

The above property may now be applied in finding the 
intersection of the two surfaces of revolution whose axes in- 
/^ tersect, as at c in Fig. 124. The simplest lines that can be 
drawn upon either surface are circles ; but if transverse sec- 
tions are made at random, there is no certainty that the circle 
thus cut from one surface will intersect the one cut from the 
other. In order to ensure this, take c as the centre of a 
sphere, — for instance, the one whose outline is ab. This cuts 
from the inclined surface the circle seen edgewise as oc ; it 
cuts from the vertical one a horizontal circle through ;;/, and 
the intersection p clearly represents a chord common to these two circles, perpendicular to 
the paper : therefore / is in this view a visible point lying on both surfaces, and therefore in 
the curve sought. 

In this case the same sphere serves to locate another point, since ab cuts the contour of 
the vertical surface in n, a horizontal through which cuts oe in r, which also lies upon the 
required curve. In like manner another sphere whose outline x?, gh, cutting the contour-lines 
in k and /, enables us to locate the point s ; and by repeating this simple operation a suiificient 
number of times, the intersection of the surfaces may be drawn with great accuracy. 

In the ordinary practice of mechanical drawing it is very rarel}^ necessary to construct 
tiie top view of an intersection of this nature. Should the occasion arise, howev^er, the 
student should have no difficulty in doing it: since the radii of the circles through ;;/, «, etc., 
are known, it will be seen that the explanations given in connection with Fig. 120 cover the 
whole ground. 

198. In Fig. 125, V and w are the vertices of two cones of revolution, whose axes, both 
of which are parallel to the paper in the front view, intersect at a. The points/" and ^ of the 
curve of intersection are seen by inspection of this view; and other points might be found by 
means of auxiliary spheres, having a for a centre, and cutting circles from each cone, as in 
the preceding example ; but if it is proposed to develop the cones, another method may be 
preferable. 




INTERSECTIONS AND DEVELOPMENTS. 



1 1 



A plane through the vertex of either cone will cut two elements from it if it cuts it at all ; 
if the same plane passes through both vertices and cuts both cones, the lines cut from one 
cone will intersect those cut from the other: but any plane through t' and w must contain 
the line vw, upon which the plane turns like a door upon the centre-line of its hinges. 

In the application of this it is desirable that the cones should have a common base. The 
base of the vertical cone is the horizontal plane represented by XN, and is a circle whose 
diameter is xy. Produce the inclined cone until NN cuts it : the section, or in other words 
the base of this cone in the same plane, will be an ellipse, whose major axis is Ik, and this is 




Fig. 124. 

readily drawn as in Fig. 105. Produce vw until it pierces the plane NN in the point/. The 
perspective sketch Z shows a cone whose base is in the plane NN, and a line vp through the 
vertex, piercing the same plane at / ; if in that plane any line/z be drawn, cutting the base 
of the cone at r and ;/, it will at once be seen that vp and // determine a plane which cuts 
from the cone the two elements vr, vti ; also, ii pj is drawn tangent to the base at t, the plane 
ipj is tangent to the cone along the element vL 

199. Now, returning to the principal diagram, draw in the top view the line /;/, cuttino 
the elliptical base of the inclined cone in r and n : this determines, as just seen, two elements, 
rzv and Jizi'. But pn cuts the circular base of the vertical cone in 0, determining in like 
manner the element vo, which cuts rzv in c and nw in </, and these are two points in the 
curve of intersection. 

A limiting point in this curve is determined by drawing in the top view pt tangent to 
the ellipse, cutting the circular base at s. Drawing the elements fzi\ sv, they intersect at ^. 
at which point sv is tangent to the curve, which is clearly shown in the front view: in order 



114 



MECHANICAL DRAWING. 




INTERSECTIONS AND DEVELOPMENTS. 



"5 



to avoid confusion, the curve is not drawn in the top view, the object being to exhibit as 
clearly as may be the mode by which the points are located. 

The process of development, and the use of such limiting lines as vs, will be readily 
understood by referring to the explanation of Fig. 120; it being hardly necessary to add, 
that a right section of the inclined cone must be made at some convenient point upon its 
axis, which will develop into a circular arc. 

200. Fig. 126 represents the hub of a screw-propeller, having the form of a sphere, with 
portions cut off by planes perpendicular to the axis at equal distances from the centre : 
these planes also limit the edges of the blade, ad and cc. The surface of this blade (here 
supposed to have no sensible thickness) is the same as that of the square-threaded screw, 




Fig. 126. 

described in Chap. V : that is, it is generated by a straight line do perpendicular to the axis, 
which, while travelling uniformly in the direction of, also revolves uniformly about the axis 
as indicated by the radial lines in the end view, the point d describing the helical arc da. 
It is now required to find the intersection of this surface with the hub. 

If do advanced along the axis without revolving, it would, when in the position 3 3, 
pierce the sphere at /i in the front view ; and 7/3 is the actual distance from the axis at which 
it does pierce it. But it has meantime revolved to the radial position 3 3 in the end view ; 
therefore set off from the centre on this radius a distance equal to 7^8, which locates the 



ii6 



MECHANICAL DRAWING. 



point r: and in like manner any number of points on the intersection ^r«^ may be deter- 
mined. 

Which may be expressed by saying that a plane perpendicular to the axis cuts a right 
line from the blade and a circle from the hub, and the circle cuts the line in a point of the 
required curve. 

If the hub be turned a quarter round toward the right, the curve due appears in the 
position /(7^; this must be first drawn in the end view, with the radii as shown : the side view 
is then determined by projecting/", which is the new position of d, to my ; s, the new position 
of r, to Ji2> ; and so on. 




Fig. 127. 

201. The blade of a propeller must have sensible thickness ; and in forming the back, it 
is common to assume the curve of intersection with the hub, which determines the thickness 
of the blade at the root. In Fig. 127, the hub, and the intersection/^^ with the acting face, 
are the same as in Fig. 126; and /r^^ is the assumed line of junction of the back with the 
hub, in the front view. To construct the end view, make transverse sections 11, 2 2, etc.: 
these appear in the end view as circles, to which the points c, e, d, and so on, are projected 
from the front view. 

Turning the hub to the right through an angle of 90^, the curve appears as hbikl \\\ the 
end view, and the front view is constructed from this by projecting the various points back 
to the planes 1 1, 2 2, as in the preceding example. 

It may be noted, that by making the sections equidistant from in the front view, as 
for instance 1 1, 1' 1', each pair will be represented by one circle in the end view, which 
thus serves for the location of two points : thus c and d We upon the circular arc 11, and so 
on. This expedient can be employed, of course, only wlien, as in the example here selected, 
the outline of the hub is symmetrical with respect to a transverse plane, — which is not by any 
means always the case in practice. 



ISOMETRICAL DRAWING. n? 



CHAPTER VII. 
ISOMETRICAL DRAWING, CAVALIER PROJECTION, AND PSEUDO-PERSPECTIVE. 

ISOMETRY. 

202. In Fig. 128, C is a top view of a cube so placed that in the front view A the 
diagonals eg, ab of its upper face are respectively parallel and perpendicular to the paper. 
The cube is shown as cut by a plane //, perpendicular to the paper in view A ; the section 
thus made, as seen in the perspective sketch V, is bounded by the three face diagonals ab, ad, 
bd: it is, then, an equilateral triangle, to the plane of which the three equal edges ca, cb, cd 
are equally inclined. And as seen in view A, this plane is perpendicular to the body diagonal 
ch of the cube, which pierces it at 0. 

In the view D, which is an orthographic projection upon the plane //, the three face 
diagonals are seen in their true lengths, forming the equilateral triangle a'b'd'. Since the 
three edges which meet at c are equally inclined to the plane, they will be equally fore- 
shortened : therefore c' is the centre of the triangle, a'c' , b'c', d'c' are equal to each other, 
and the three angles at c' are each equal to 120°. 

Every other edge of the cube being equal and parallel to one of these three, each visible 
one will appear equal and parallel to one of those already drawn ; thus the apparent contour 
of the entire cube will be a regular hexagon, the representation of each face being a 
rhombus. 

Because the edges of the cube are thus foreshortened in the same proportion, so that 
they and all parallels to them may be measured by the same scale, such a view as D is called 
an Isometric Projection ; a'c , b'c' , d'c are called the isometric axes; the planes which they 
determine, and all planes parallel to them, are called isometric planes; and all lines parallel to 
the axes are called isometric lines. 

203. Drawings made in this manner possess the advantage of conveying, in one view, 
ideas of the three dimensions, as do those made in perspective ; and in many cases they 
exhibit the peculiarities of structure more clearly than ordinary plans, sections, and elevations. 
They are readily understood by those who are not familiar with common projections ; and in 
making sketches this system is very useful. 

Obviously, however, the advantages of isometry are more pronounced when the objects 
to be represented are bounded by right lines, of which the principal ones are parallel and 
perpendicular to each other. It is not well adapted for the general drawing of machinery, 
since it involves an unpleasant distortion, and also because in most cases the circles are 
projected as ellipses. 

204. Distinction betweeti isometrical projection and isometrical drawing. — In Fig. 128 
the actual length of the edge of the cube is cd\ its apparent length in view D is c' d' , equal to 
odm view A. Suppose cd to be one unit in length — an inch for example: then by taking od 



ii8 



MECHANICAL DRAWING. 



as a unit it is possible to construct an isometric scale, by which all the isometric lines in D 
might have been set off; and such a scale could be used in constructing any isometrical 
projection. 

This is a matter of purely abstract, theoretical interest, and not of any practical use 
whatever. Since the isometric lines are all equally foreshortened, there is no reason why they 
should be represented as foreshortened at all. Consequently an Isometric Drawing of the 
given cube is made as shown at E, each edge being drawn of its true length. This is the 
method always adopted in practice, the scales in common use being alone employed. The 
man who should construct a true projection, and send it to the workman to be measured, by 




m 

e a 



Tip^.128 



an isometric scale, would simply make a record of his own stupidity; he who should teach 
others to do so, would commit a blunder of much more serious importance. For, to use the 
words of another, " the value of isometry as a practical art lies in the applicability of common 
and known scales to the isometric lines."* 

205. We proceed, then, just as in making ordinary working drawings, setting off the 
dimensions on those lines either " full size," or with the 3-inch scale, the i^-inch scale, etc., 
as the case may require. Naturally, lines which are vertical are so represented ; the other 



*W. E. Wonhen. 



ISOMETRICAL DRA WING. 



119 



isometric lines are then drawn with great facihty by the aid of the T-square and the triangle 
of 60'' and 30°. 

Figs. 129-134 are simple exercises, composed wholly of isometric lines, the construction 
being so obvious that no detailed explanation is required : the method of locating the foot of 
the cross in Fig. 132, and the mortise and the tenon in Fig. 134, by measuring along the lines 
ca, cb, or parallels to them, is sufficiently shown by the dotted lines. 

It is to be distinctly understood that these figures are illustrations merelj-: the student 
is not to copy them, but to construct them or others of similar character, with such variations 
of dimensions, arrangement, or design as may be suggested by his ingenuity, which should be 
given full play. 

In regard to the shadow-lines, reference is made to a cube as a standard. Thus in E, 
Fig. 128, the light is supposed to have the direction of the body diagonal af; consequently 




Tio\134;. 



the faces ec, eg are illuminated, and shadows are cast by the edges ed, dc, cb, and bg. In 
drawing, the first of these lines should be made the heaviest, the last one the lightest, and the 
other two of equal and medium thickness. 

206. Isometrical Drawing of the Circle. — The ellipse representing the circle inscribed 
in the face of the cube, Fig. 135, might be drawn by the method of Fig. 40 (Chap. Ill But 
the axes coincide with the diagonals, and are at once determined by representing the parallels 
through /, ill, n, 0, in the elevation shown at the left. Describe a semicircle upon cd as a 
diameter, divide it into four equal parts by the points 1, 2, 3, draw 3 3', 11 perpendicular 



I20 



MECHANICAL DRAWING. 



to (:</, and through 3' and 1' draw parallels to bc\ these will cut the diagonals at w, o, and 
n, I, thus limiting the major and minor axes. As a check, note that Im and on should be 
parallel to cd. 

The sides of the rhombus being equal, this construction may be made upon either one at 
pleasure. And, since all the faces of the cube are exactly alike, it follows that all circles lying 
in isometric planes are represented by similar ellipses. 



Tig-. 135 




By drawing tangents at the points /, in, n, o in the elevation the circle is circumscribed 
by a regular octagon, the isometric representation of which is therefore made by drawing at 
the corresponding points /', ni , ?i\ o' , in the upper face of the cube, perpendiculars to the 
diagonals, terminating in the sides of the rhombus. 

207. Graduation of the Isometric Circle. — First Method. At the middle point d oi gb, 
in Fig. 136, erect a perpendicular de equal to db, and draw cb, eg; about ^ as a centre 




describe with any radius the quadrant rs, divide it as desired by the points 1, 2, 3, etc., 
through which draw radii and produce them to cut gb. From these intersections yNith gb 
draw lines to/, the centre of the ellipse: these will cut its circumference in the required 
points of division, 1, 2, etc. 



ISOMETRICAL DRAWING. 121 

Second Method. Describe a semicircle upon the major axis wo as a diameter, divide it 
in the desired manner by the points 1', 2', etc., through which draw perpendiculars to mo, 
cutting the circumference of the ellipse in 1", 2", etc. : these will be the points required. 

An application of the above is shown in the drawing of the bolt, nut, and washer, Fig. 
137. About/, the centre of the outer ellipse, describe an arc with radius />c? = semi-major 




Ficr.137. 



axis, set off the arc oa = 60°, erect the vertical ad, and draw 1>J> cutting the inner ellipse 
(circumscribing the base of the nut) in c. 

208. To draw Angles to the Sides of the Isometrical Cube. (Fig. 138.) — Draw a 
square c£^, whose side is equal to the edge of the cube ; about one of its angles, say c, as a 
centre, describe the quadrant ai, graduate it, and produce the radii through the points of 
division to cut the sides of the square. The scale of tangents thus formed may, by cutting 
out the square, be applied to any side of the isometrical cube, thus determining the direction 




^ 3i 



Fig. 



138. 




of a line in its face which shall represent a line making any required angle with its edge. 
For example, make a'e' = ae, and b'f — bf: then a'c'e' , b'c'f are the isometrical representa- 



122 



MECHANICAL DRAWING. 



tions of the angles ace, bcf. The same angles are represented on the left-hand vertical face of 
the cube by making d' e" = ae, c'f" = bf, and drawing a' e" , a'f". 

An application of this is found in making the isometrical drawing of the piece shown 
in plan and elevation at C and A, Fig. 139, in which the angles x, y are assigned: also in C 
the distance ad is given, dke is perpendicular to ad, and cf parallel to ak : the thickness is 
uniform, and equal to aJi in view A. The isometric drawing is lettered to correspond, and 
should require no further explanation. 

209. Another method of dealing with lines which, though lying in isometric planes, are 
not parallel to either of the isometric axes is by means of " offsets." Thus in Fig. 140 
the slope of the diagonal brace is determined by measuring the distances cd, de along the 
isometric line ca, and setting up the vertical ef, of the values ascertained from the elevation 




shown on a reduced scale. This really amounts to the same thing, the angle being 
constructed by laying off the base and altitude of a triangle of which the required line 
is the hypothenuse, — which is in perhaps the majority of cases the most convenient means. 
Another illustration is given in the drawing of the box. Fig. 141 ; the outline of the end 
of the partially opened lid being set out by means of the vertical measurements ca, co, cb, ch, 
and the offsets ac, ol, bd, kg, taken directly from the transverse section shown at the left. 

210. This principle may be extended, and is applied to the determination of lines which 
do not lie in isometric planes; as illustrated in Fig, 142, representing the roof of a cottage, 
of the form and proportions shown in plan and elevations on a smaller scale at the right. 



ISOMETRICAL DRAiVING. 



12: 



The sloping lines of the roof at the nearer end are found by setting up the heights ab, ad on 
the vertical through a, and drawing the isometric lines bh, df : then the points z, k are the 
intersections of bf, fh with the isometrical line through c. A similar construction may be 
made at the farther end, thus fixing the line of the ridge//"', on which the pointy is located 
by setting o^ fg, its distance from the plane de: we can then diX2i\\ gi, gk, which do not lie 
in any isometric plane. 

The same process is applied in drawing the wing roof, the heights n, v, r being set up 
on the vertical through the nearer corner w, and the distance oq measured from the plane 
mo. The ridge line will pierce the main roof at a point ?/, which may be thus located : Set 
up at = mr, draw the isometric line ts cutting bf in s, and through s draw a parallel to ff: 
this will cut the ridge line of the wing in the required point. 




Fip-.142. 



It will be observed that the lines fh, gk, and ff, differ very little in direction, and 
qz, OH differ still less. This simply shows that the farther side of the main roof is very 
nearly, and that of the wing roof almost exactly, perpendicular to the plane upon which 
the isometric drawing is made; and it will be perceived that in such cases this is not a 
peculiarly eligible mode of representation, — as indeed it is not for architectural subjects of 
any description. 

211. Thus far one of the isometric axes has been made vertical. But inasmuch as it is 
the relative direction of the lines among themselves which determines whether a drawing 
is an isometric one or not, there is no necessity that any of them should be vertical. In 
Fig. 143, for example, the principal lines are horizontal ; but the drawings of the die and its 



124 



MECHANICAL DRAWING. 



matrix, and of the timber with its mortises and its tenon, are at once recognized as isometric, 
and are just as easily understood as if they stood upright. 







Fig. 143. 

For convenience in constructing the drawings by means of the T-square and triangles, it 
is preferable in most cases, of course, that one of the isometric axes should be either vertical 
or horizontal, but should there be any reason for selecting other positions, there is nothing 
in the principle of isometry to prevent their adoption. 




Fig". 144. 




212. It will be noted that the correspondence of the die to the matrix in Fig. 143 is made 
mucli more obvious than it otherwise would be, by exhibiting the opposite ends of the two 
pieces. By merely turning the page around, it will be apparent that this could have been 
done equally well if the two pieces had been drawn in a vertical position. 

For this reason isometry affords a means of illustrating in a very clear and striking man- 
ner many subjects in which views of the lower surfaces are desirable : a good example is 
shown in the drawing of the small shelf with its supporting bracket. Fig. 144. 

In making such a drawing, as will readily be seen," the process is equivalent to construct- 
ing the projection of the cube, Fig. 128, upon the plane //, as seen from the lower left-hand 
side, and looking in the direction opposite to that indicated by the arrow. 



ISOMETRICAL DRAWING. 



125 



213. In Fig. 145 is shown an isometric drawing of the ratchet-wheel represented in the 
full size views at the right. The backs of the teeth not only terminate in, but are tangent to, 
the interior circle ; and a test of the accuracy of the isometric drawing is found in the 
tangency of the edges to the ellipse representing that circle. And this embodies a principle 
capable of many other applications — as, for instance, in laying out a wheel with radial taper- 
ing arms : the side outlines of each arm are tangent to a circle, which being drawn in the 
isometric construction, it is seen that the breadths of the arms at the outer ends only need be 
set out, thus fixing points through which tangents are to be drawn to the ellipse. 

214. It seems needless to multiply examples, as it is believed that by the aid of the pre- 
ceding any isometrical drawing likely to be required in practice may be constructed. 




In drawings of machinery, the circles of wheels, bearings, ends of shafts, and the like, will 
usually lie in isometric planes. Should occasion arise to represent one which does not, 
circumscribe it by a square : the projection of this will be a parallelogram, within which the 
ellipse may be drawn by methods previously explained. So, too, if it should be necessary to 
represent the section of a cylinder or a cone by an oblique plane, the solid may be conceived 
as surrounded by a square pyramid or prism, Avhose section by the given plane, as well as the 
isometric drawing of it, will be a parallelogram circumscribing the required ellipse. 

215. In conclusion, it may be pointed out that many lines not usually classed as isometric 
are strictly so in fact. This distinctive term is technically restricted to lines parallel to the 
isometric axes, which again are so called because they are equally foreshortened, and this is 
the result of their equal inclination to the plane upon which they are projected. Now, in 
Fig. 146 the cube is cut by the plane// as in Fig. 128, and in the projection D we at once 
recognize c'a' , c'b\ c'd' as the isomictric axes. If «:</ revolve around co as an axis it will gener- 
ate a cone dcli, all of whose elements make the same angle with the plane//; so that any one 
of them, as cr (seen in D as cV), would be foreshortened in the same proportion as any other 
one. But in the isometric projection this fact would not be indicated by merely drawing c'r' : 
it is necessary to locate the point r' hx means of offsets — c'n giving its distance from the 
plane a'c'd\ c I its distance from the plane a'c'b' , and mn its distance from the plane b'c'd'. 



126 



MECHANICAL DRAWING. 



This being done, / is at once seen to be the foot of the perpendicular r' s' from the point in 
question to the plane last mentioned. 

Again, the isometric projection of any frustum of a cone, xyuv, whose bases are parallel 
to //, would appear simply as two concentric circles, and without some auxiliary view that 
projection would convey no definite information about the cone, which might be of any alti- 
tude or have either base uppermost. 

Since the whole value of isometry, in practice, lies in the power of imparting in one view 



Tio-.i^e 




definite ideas of the three dimensions, the above hints may serve a purpose as indicating 
possible relations of parts for the representation of which this method of drawing is not 
suitable. 

CAVALIER PROJECTION. 

216. In Fig. 147, let Vl/TV represent a vertical glass plate (corresponding to one side of the 
show-case, Fig. 68) ; let r be a point in this plane, and ca a line perpendicular to it. Let ar be 
a visual ray, making an angle of 45° with the plane MN, and piercing it at/: then cp is the 
representation of ca upon the picture pbne, and it is equal to ca, because the angles cpa, cap 
are each equal to 45°. 

Suppose the eye to be at an infinite distance in the direction ar: then all the visual rays 



CAVALIER PROJECTIONS. 



127 



will be parallel, and all lines perpendicular to MN will be represented upon that plane by 
lines of their actual length, and parallel to cp. 

The fact that the projection is of the same length as the perpendicular line depends upon 
the condition that the picture plane cuts the projecting lines at an angle of 45°. But the 
direction of the projection depends upon that of the visual ray. Thus if the eye be still at 




an infinite distance, but in the direction at, the projection will have the direction co, but its 
length will remain equal to ca. 

Thus the direction of the projecting lines may be parallel to any element of the cone 
whose axis is ca, the angle at the vertex a being 90°, since all these elements make angles of 
45° with the picture plane MN. 

217. Now any line which lies in the picture plane is its own projection. In representing 
a cube, therefore, as in Fig. 148, we may assume its nearer face to lie in that plane, and it 
will thus appear of its true form and size, that is, a square, as shown. From the preceding it 
follows at once that the edges which are perpendicular to MN may be represented by parallel 






Tig-. 148. 

lines of their true length, but having any direction at pleasure ; which enables us to show, in 
addition to the front face, either the right face or the left, the upper or the lower, as may 
best suit our purpose. And, as the figure shows, either of these faces at will may be made 
more conspicuous than the other by proper selection of the angles. 

We have, then, a system of true oblique projection : it is more flexible than the isometric, 



128 



MECHANICAL DRAWING. 



always quite as easily executed and in many cases more so, and like it exhibits the three 
dimensions in one view. All lines lying in planes parallel to the paper are shown in their true 
forms and relations ; and not only these but lines perpendicular to the paper are shown of 
their actual dimensions, the introduction of any such senseless appliance as the " isometric 
scale " being prevented by the very nature of the process. 

218. This system is well adapted for purposes similar to those in which isometric drawing 



Pip-.i^a 




is employed, — such as the representation of joiner-work, as exemplified in the case of the box. 
Fig. 149, and in that of the peculiarly notched and fitted pieces shown in Fig. 150. In the 





\ 


^ 


^ 






^ 


/""^ ; 


; 












^ 


■ ■■T" 


L' 


-^-^ 




^ 












Y 













Fig-.i50. 




illustration, and especially in the sketching of small mechanical details, it possesses the 
decided advantage over isometry that, as shown in Fig. 151, circles whose planes are parallel 
to the paper are represented by circles, which greatly expedites the work of construction. 



CA VALIER PRO JECTIONS. 



129 



Those lying in planes perpendicular to the paper, however, must here too be represented by 
ellipses : since each circumscribing square is projected as a rhombus, the axes will coincide 
with the diagonals, and may be found as in Fig. 135. 




The use of ordinates, or offsets, in determining lines which are neither parallel nor per- 
pendicular to the paper is substantially the same as in isometric drawing. Thus in Fig. 152 







the point/ in the plane eg is located by setting off gh, its distance from the plane gn, and 
then hf, its distance from the front plane gm ; the point k is determined by ab, its distance 
from the end plane ^«, bd its height above the plane eg, and dk its distance in front of the 
rear plane, which is invisible. The two points/ and k being thus fixed, the projection of the 
line fk is determined ; and the rest of the construction can be readilj- traced without expla- 
nation. 

The lines which cast shadows, and are therefore to be made heavj', can usually be 
determined by inspection, — the light, as in the common orthographic projections, being sup- 
posed to come from over the left shoulder, and to go downward to the right as it recedes, in 
the direction of the body diagonal of a cubical room upon the wall of which the paper is 
hung, in front of the observer. 

219. There is, then, no need to pursue this subject farther : the principles which have 
been thus briefly set forth are sufificient for applying either of these modes of projection to 
any subjects within the common range of practice. Both are very useful, with certain limita- 
tions which have been suggested, and the question whether either is suitable for any given 
case can be settled by experience alone. 



130 MECHANICAL DRAWING. 

But one thing has been decided by experience beyond all question ; and that is, that the 
attempt to apply either isometric or cavalier drawing in the construction of a general plan of 
any complex machine is certain to result in a melancholy failure : the distortion, less notice- 
able in the case of minor details and detached pieces, becomes unendurable when the various 
parts are assembled. 

PSEUDO-PERSPECTIVE. 
220. For the purpose of producing a certain effect of relief, and conveying at least some 
idea of the three dimensions, while at the same time avoiding this distortion as well as the 
labor of constructing a true perspective drawing, a mode of representation has been devised, — 
to which the name of Pseudo-Perspective seems appropriate, — of which we give a single illus- 
tration in Fig. 153. 













f"\ 


/ 
















\ 

\ 



Fig. 153. 

This is, in principle, a modification of cavalier projection ; in that, as has been stated, 
the parallel projecting lines are inclined to the picture plane at an angle of 45°. But, refer- 
ring to Fig. 147, it will be seen that if the cone of visual rays should have a less angle at the 
vertex, the projection of ca would be shorter; and by properly choosing this angle, the pro- 
jection may be made shorter than the line in any desired ratio, while its direction is still 
entirely arbitrary. 

The pseudo-perspective drawing, then, is made by representing the lines which are 
parallel to the paper in their true size, while those which are perpendicular to it are reduced 
to, say, one twelfth of the actual length, but parallel. This of course renders the result value- 
less as a working drawing ; but it gives a sense of depth, and such representations are in many 
cases well suited for illustrations upon a small scale, such as cuts for encyclopsedias and the 
like. 

The distortion in the true cavalier projection is due to the mental recognition of the 
facts that the true representations of receding lines ought to converge, and that equal dis- 
tances upon them ought to appear less as they recede. Both these errors are made less 
conspicuous by reducing the lengths of these representations, which is accomplished by the 
method of drawing above explained. 



THE SPUR-WHEEL AND THE BEVEL-WHEEL IN INCLINED POSITIONS. 131 



CHAPTER VIII. 

THE SPUR-WHEEL AND THE REVEL- WHEEL IN INCLINED POSITIONS. CONSTRUC- 
TION OF THE CLOSE-FITTING WORM AND WHEEL. CONSTRUCTION OF THE 
SCREW-PROPELLER. STANDARD SECTIONING, ETC. 

221. It is assumed that the student has now gained a knowledge of principles and 
modes of representation, and a degree of skill in the use of instruments, which will enable him 
to make presentable and accurate working drawings of mechanical details. But it is not 
proposed to give, as might be done, a collection of such drawings to be repeated as mere 
exercises. 

Familiarity with the construction, and positive apprehension of the proportions, of any 
machine is best gained by actual examination of it ; the copying of plans will never give an 
adequate idea of how the pieces would look when finished in accordance with them. The 
novice in designing is often much astonished when he sees in the solid the first piece for 
which he has made the drawing : he recognizes the form, of course, but it looks too large or 
too small, this part seems too light or that too heavy ; and on the whole the physical appear- 
ance is somehow different from his mental conception of it. The fact is, let him who thinks 
he can, account for it as best he may, that there is a mysterious something in the relation 
between the drawing and the structure, the power to grasp which at once is born in few ; and 
those who have it not by nature, will acquire it but slowly and painfully if they confine them- 
selves to paper. 

222. The intention therefore is, that the student should at once begin to acquire it in 
a more logical and practical manner. Instead of copying a plan, let him make measurements 
and sketches of some simple machine or part of a machine already built, and lay out a working 
drawing from the data thus obtained. Being thus obliged to take special note of the shape, 
size, and weight of the object in question, the impression upon his mind will be more distinct 
and lasting, and his drawing when done will convey to him a fuller meaning, than if the 
opposite course had been adopted ; and besides, practice is had in a most essential acquirement. 

The first efforts in this direction should be confined to com.paratively simple subjects, of 
such dimensions that the drawings can be made of full size : the immediate object of this 
training is not to acquire skill in the construction of complex and difficult drawings, but to 
gain the power of forming, by inspection of any working plan, a correct conception of what it 
means. And experience has amply proved that this is best done by dealing at first with 
drawings of the same size as the objects themselves : whatever the reason may be, some little 
practice is usually needed in order to realize correctly in the mental image, the dimensions 
indicated by plans upon a reduced scale. Nature never acts per saltnni, and this practice 
should progress by easy gradations. 

223. Those who are left to instruct themselves must perforce devise for themselves the 
ways and means of making the measurements required ; and if they succeed unaided, so much 



132 MECHANICAL DRAWING. 

the better for them, — just as it is better for a child thrown into the water to swim than to 
drown. But the best swimming-masters have decided that it does not pay to pitch apphcants 
overboard and select classes from the survivors. Exactly so here : it is not the sole duty of 
an instructor, if there be one, to say " Sketch this," " Draw that," and then expect of his pupil 
a perfect result. If he knows how to show him how, it devolves upon him to do so if appealed 
to ; and if the services of a thoroughly competent and experienced teacher are of value at any 
stage of progress, this is emphatically the one at which their value would be the greatest. 
And this not in regard to the sketching alone, but in the way of suggesting the best selection, 
and most important of all, the best arrangement, of the'views which should be made in the 
working drawing. 

In the second part of this work the matter of sketching is treated more at length, and 
hints in regard to arrangement will also be found in connection with the illustrations of that 
and other subjects. 

We proceed, then, to the consideration of a few rather miscellaneous topics which may 
subsequently be found of service. , 

224. Drawing of a Spur-wheel in an Inclined Position.— In Fig. 154 we have an end 
view, a top view, and a section of a spur-wheel. Referring to Fig. 55 in relation to the forms 
of the teeth, it is sufficient to say that if the interior describing circle be of half the diameter 
of the pitch-circle the flanks will be radial lines, as here shown : the manner of drawing the 
top view and the section in this figure should require no explanation. Fig. 155 is a view of 
the same wheel, not in the direction of the axis as in Fig. 154, but at an angle with it. It is 
at once perceived that the pitch-circle WW will be projected here as an ellipse ; and the 
points such as c, d, at which the teeth cut the pitch-circle, are projected to c' , d' in the top 
view in Fig. 154, and thence to c'\ d" , upon the ellipse W"W" in Fig. 155. 

Similarly, the tops of the teeth lie upon the outer circle of the wheel-blank, which also is 
seen as an ellipse in the inclined view ; and so a, b are projected first to a' , b' , in Fig. 154, and 
thence to a", b" upon the corresponding outer ellipse. 

If any circles be drawn between the pitch-circle and the outside one, they also will be 
seen as ellipses in Fig. 155, and by treating as above the points in which they cut the outlines 
of the teeth, as many intermediate points may be determined. 

A like proceeding would enable us to draw the contours of the flanks were they curved as 
in Fig. 55. For the sake of convenience they are here made radial; for which reason they 
will in Fig. 155 appear as right lines converging to the centre of the ellipse W"W". 

All of the above relates to the appearance of the nearer side, or rather end, of the wheel. 
In regard to the farther one, it is clear that a pitch-circle scribed upon it would also be seen 
in Fig. 155 as an ellipse similar to W"W", of which a portion is shown at Y"Y"; and to 
this all the preceding applies in a way too obvious to call for explanation. The consequence 
is that the visible edges of the teeth, such as those through a" and b", and the visible tangent 
elements, such as that through x, will be straight lines, of equal length, and of course parallel 
to the axis. 

225. Drawing of a Bevel-wheel. — The basis of the bevel-wheel is a cone, vab, in the 
top view. Fig. 156. In the end view the base ab appears as a circle, which is called ^\\q pitch- 
circle of the wheel. The first step is to divide this circle into as many equal parts as the 



THE SPUR-WHEEL AND THE BEVEL- WHEEL IN INCLINED POSITIONS. 133 

wheel is to have teeth : this determines the pitcJi, measured by the arc gi, and the thickness 
^is a Httle less than half the pitch. 

Now draw at « and b lines perpendicular to rtz* and bv; these meet the axis at 11, and 
form the outline of a cone nab, normal to the pitch-cone. If this normal cone be developed, 
its base will become a portion of the circle whose centre is ;/ and radius nb ; an arc of which, 
bgf, is shown in the top view. Draw a radius of this arc at any point as 0, on each side of 
which set off an arc, ^/, og, equal to half the arc ^ in the end view. (Practically, it suffices to 




bisect^/ and set off that distance on each side of 0, because, unless the tooth is of unusual 
thickness, the difference between the arc and its chord is inappreciable.) Construct and 
proportion the outline of a tooth, regarding bgf d.?> an arc of the pitch-circle of a spur-wheel. 
If this developed tooth were cut out of thin metal, and wrapped back upon the normal cone, 
its contour would be that required for the tooth of the proposed bevel-wheel. The doing of 



134 



MECHANICAL DRA WING. 



this graphically, it will be seen, is an application of the process explained in connection with 
Fig. 105. 

226. Having determined the height rs of the tooth, make br, bs, in the top view and in 
the section, respectively equal to or, os : then the tops and the bottoms of the teeth will be 
limited by the circles described by 5 and r in revolving around the axis. 




rig\i57. 



Next decide upon the length be of the tooth in the sectional view, and draw at c a per- 
pendicular to be, also rk, st, converging to the vertex of the pitch-cone. It will be seen that 
the inner ends of the teeth lie upon another normal cone, edm, in the top view, and that they 
will be of the same form as the outer ends, but smaller in the proportion of tk to sr. 

The radius of the circle described byjisjA'; upon this circle in the end view make />^ 
equal to Ih in the development: then the foreshortened view of the face curves will hQfh,gl\ 



THE SPUR-WHEEL AXD THE BEVEL-WHEEL IX IXCLIXED POSITIOXS. 135 

intermediate points in these curves may be found by means of intermediate circles, as indi- 
cated. The same method might be adopted for drawing the flanks, if they were curved ; but 
since they are in this case made radial in the development, they will appear radial in the end 
view, and the apparent depth of the spaces between the teeth is limited by the circle whose 
radius is ry in the section. 

The tops of the inner ends of the teeth lie upon the circle whose radius is tz, and their 
breadths may be determined by drawing radial lines as through / and //, the points limiting 
the breadth at the outer end ; and these lines represent the edges of the teeth. So, too, 
radial lines through f and g cutting the inner pitch-circle whose diameter is cd, locate the 
junctions of the faces and flanks ; since the latter in this instance are straight, they are also 
radial, and terminate in the circle whose radius is kii in the section. 

227. One tooth having been laid out, all the others are copied from it in their proper 
positions in the end view. This is a necessary preliminary to the construction of the top view. 
In this, all the circles mentioned appear as straight lines, to which the points in which the 
tooth outlines cut these circles are projected from the end view, as shown at / and ^ on ab, h 
and /on the outer circle through s. Since the teeth are here made with radial flanks, the 
outHnes of these will be straight, those upon the outer normal cone converging to its vertex «, 
and those which are visible at the inner ends of the teeth converging to m, the vertex of the 
inner normal cone. 

It is thus quite evident that the correct drawing of the bevel-wheel is, if not a compli- 
cated, nevertheless quite a tedious, operation. Fortunately, this labor is not necessarj' for the 
purpose of simph- having the wheel made: the shop drawing (Fig. 17, Part II) is far more 
easil}^ made. But in a general plan of a complete machine, or for illustrative purposes, it 
may be necessar}' to draw such exterior views. 

228. Drawing of a Bevel-wheel in an Inclined Position. — This involves no small 
amount of additional work, as will readily be seen by study of Fig. 157. But few words of 
explanation, however, are necessarj'-, since the mode of operation is substantially the same as 
in Fig. 155. All the circles are now projected as ellipses, to which the various points of the 
tooth-outlines, as^", h" , I", are projected from the top view of Fig. 156. The vertices of the 
three cones appear in this new view as v" , in" , n" upon the axis, and to these points the edges 
of the teeth, the flanks at the inner and those at the outer end, respectively converge. 

229. Drawing of a Worm and Wheel. — The form of this device selected for illustration 
is that commonly called the close-fitting tangent screw : a steel screw of the same size as that 
finally to be used is notched to make a cutter, which is then hardened, tempered, and used 
for cutting the teeth of the worm-wheel. In this operation the cutter can remove only just 
so much metal as to permit of rotation : the ultimate fit of the driving worm is therefore 
as perfect as it can be made. 

In Fig. 158 the diagram on the left is a section through the axis of the worm and 
perpendicular to that of the wheel ; that is to say, by a plane LL in the diagram on the 
right, which is a section of the wheel by a plane passing through its own axis, and perpen- 
dicular to that of the screw. 

Let C be the centre of the wheel and WPr a portion of its pitch-circle, to which draw 
TP tangent at P\ and regarding this as the pitch-line, the next step is to lay out teeth as for 



136 



MECHANICAL DRAWING. 



a rack and wheel : the outline of the rack will be the meridian section of the worm. The 
outlines of the rack-teeth might be composed of cycloids or other curves, but clearly it is 
easier to make a screw with a section bounded by right lines. 

230. Fortunately, it is also better; and it may be done as follows: Draw the right line 




ricr.158. 



API, making an angle of from 15° to 17° with TP\ let fall upon this the perpendicular CI, 
with which as radius describe about C the dotted circle shown. The sides of the wheel-teeth 
are involutes of this base-circle, and one side of the rack-tooth is perpendicular to AI\ the 
other side of course being equally inclined in the opposite direction. 



DRAWING OF A WORM AND WHEEL. 137 

Divide the pitch-circle into as many equal parts as there are to be teeth in the 
wheel : this determines iht pitch-arc, PD, in the diagram. On PT set off PE equal to this 
arc : then if, as here supposed, the worm is to be single-threaded, PE is the helical pitch. 
Practically, the projection of the rack-tooth below TP may be limited by a horizontal line 
either passing through / or a very little above it : this line then will be the lower outline of 
the cylindrical blank for the worm, of which the diameter may be from three to four times 
the pitch PE; when this diameter is decided the axis can be drawn; practically, also, the 
pitch-line TP may bisect the total height of the rack-tooth, which determines the bottom of 
the space and the diameter of the core of the screw. The tops of the wheel-teeth must just 
go clear of this core, and in like manner the spaces in the wheel must be deep enough to 
clear the outside of the worm: these spaces may be drawn with radial lines, within the base- 
circle. 

The worm itself, then, is merely a V-threaded screw with a peculiar angle for the sides of 
the thread, and is easily drawn as explained in Chap. V; it is here shown in section only, for 
the sake of exhibiting the construction more clearly. 

231. The fourth tooth in the diagram thus far considered shows not only the outline aec 
in the central plane LL, but also the extreme visible contour abdqp, bounding the tooth at the 
back, beyond the plane of the section. The determination of this contour requires an 
explanation of the right-hand diagram, in which O is the axis of the worm, the outer circle 
through a is the circumference of the blank, the inner circle through c is the circumference of 
the core, and the intermediate circle through e is the circumference of what may be called 
the pitch- cylinder, generated by the revolution of TP about the axis. The wheel is usually 
made, as here shown, to embrace about \ of the circumference of the worm : in this view, 
then, the lines Ob, Oi include an angle of 60°, and Oi produced cuts the axis of the wheel at 
V, and revolving about that axis generates a cone, which may be regarded as standing in the 
same relation to the worm-wheel that the normal cone had in relation to the bevel-wheel. 

In applying analogous treatment, the argument would be as follows: (9 F" represents the 
plane tangent to the cone, which cuts from the pitch-cylinder of the worm a right line, here 
seen as the point e" , and this line moves in rolling contact with the circle described by the 
point e" of the wheel, in revolving about the axis VL. This circle is also the base of the 
normal cone, which would develop upon the tangent plane into an arc of a circle whose 
radius is Ve" : regarding this as a pitch-circle, the problem now is to find the form of a tooth 
which shall work with a rack whose form is that of the section of the screw by the same 
plane OV. • 

232. Fortunately, this is easily solved, because all the sections of the screw by planes 
through its axis are alike. Therefore, produce PC, making PC = Ve", and draw CT per- 
pendicular to AT produced : it will then be seen that the involute of a base-circle whose 
radius is CT is the correct outline for the developed tooth. Or at least, if this were wrapped 
back upon the normal cone, the approximation to the required curve ^aT would be as close as- 
is attained in the ordinary mode of laying out bevel-gearing. 

But again, it is not necessary in any common case to take even this trouble, for two 
reasons : one is that this remote line is not a working line, — it is put in merely for effect and 
finish, and the cutter will make it right in metal however it is drawn ; and the other is, that 



138 



MECHANICAL DRAWING. 



this new involute will differ from the first one, within the limits used, so slightly that the dif- 
ference will be inappreciable unless the work is on a prodigious scale. Practically, then, we 
may make this curve bd s. portion of the same involute as the tooth-outline aec: we have now 
to determine its location and its limits. 

233. The worm is a rack which advances by rotation ; and one turn of it is equivalent to 




a longitudinal travel through a distance equal to the pitch. Now the angle aOb includes j^^ 
of the circumference; and in turning through this angle the worm would drive the wheel 
through Jj of the pitch-arc. This fraction of the pitch, then, is set off from e on the pitch- 
circle WPr, thus locating the point n, through which the involute is drawn for the back of the 
tooth ; whicli is limited at the top by a circle whose radius is hd, and at the bottom by one 
whose radius is gb, taken from the right-hand diagram ; in which it is also seen that a curve 



DRAWING OF A SCREW PROPELLER. 139 

must extend from c to d, and another from a to b, each of which will be a sort of helix, lying 
however upon a concavo-convex surface instead of a cylindrical one. 

An outside view of a complete tooth, greatly enlarged, is given below, where /';/' is an 
arc of the pitch-circle, and the dotted outline a'c'p'z is the section of the tooth by the plane 
LL\ e'n is y^ the pitch, b'yi'd' the remote contour of the tooth. The contour on the nearer 
side of the wheel is a similar curve i'ln'k', e'ln being equal to e'n ; and k'c'd', i'a'b' are the 
quasi-helical curves above mentioned. Points in these curves might be determined by draw- 
ing other normal cones inside of OVL, and repeating the process just described, by which k' 
and d' were located ; but for ordinary purposes of representation this is needless, the limita- 
tion being already quite narrow : thus, for instance, the upper curv^e must be tangent to p'c' 
at the point c' . 

At the opposite side of the tooth r's , r't' are also each Jg- the pitch, and the same cur\'es 
are drawn in the reverse direction, of which x't'u' alone is visible. 

234. Indeed, few of the curves above discussed are actually seen in the front view of the 
worm-wheel, as shown in Fig. 159. But it is necessary to determine them nevertheless if it is 
required to construct the side view, as will be clearly seen by aid of the letters, which cor- 
respond with those of Fig. 158. 

All the teeth must first be drawn in the end view, as was the case with the bevel-wheel : 
the points of their outlines lie upon circles which appear in the side view as straight lines, to 
which they are projected one by one from the end view. 

As in the cases of the spur-wheel and the bevel-wheel, it is wholly unnecessary for the 
purposes of construction to make these outside views of the worm-wheel, which involve so 
much labor : for illustrations of the working drawings for the shop, the reader is referred to 
Part II, Fig. 18. 

235. Drawing of a Screw-propeller. — Fig. 160 shows the hub and one blade of a screw- 
propeller, of which three views are given, viz.: on the right, a view from the starboard side of 
the vessel, at the left of which is a view from aft, looking forward, and above this is placed 
the top view. The subject chosen for illustration is a true screw, that is, one whose surface is, 
Kke that of the familiar square-threaded screw, generated by a right line all of whose points 
describe true helices of the same pitch ; and in this case it also remains always perpendicular 
to the axis: the pitch being assigned, any one of these helices is readily drawn. The 
element 00, in the central transverse plane of the hub, is assumed to be vertical : then the 
intersection aob of the acting face with the hub is determined, and the intersection awb of 
the back is assumed, as in Figs. 126 and 127. 

This being done, the blade is otherwise limited as follows : A definite distance x, aft of 
the hub, being assigned, a plane // is drawn perpendicular to the axis, which fixes the over- 
hang of the trailing edge ; also the radius of the small circle whose centre is 1 being given, an 
arc is described with that radius, tangent to // and to po the outline of the cylinder whose 
diameter is that of the screw, and ac is drawn tangent to this arc. Also, a distance od is 
assigned, and bd drawn, to fix the overhang of the leading edge ; an arc with given radius 
being now described, tangent to od and bd, the dotted line ayeozb is the contour of a box, as 
we may say, within which the propeller is required to revolve. 

236. The problem then is, to determine the intersection of the acting face with this 
peculiarly shaped surface of revolution. 



14° 



MECHANICAL DRAWING. 



Suppose the vertical element oo to advance without rotating to the position uu in the 
side view : it would then cut ac at the point v, whose distance from the axis is the same as that 
of /. But while thus advancing, the element in fact does turn, so as to assume the radial 
position im in the end view ; describe about C an arc with radius Cj, cutting this radius 7tii in 
g, which will be a point in the intersection. By repeating this process any number of points 
may be located, and the apparent contour of the blade fully determined. Since the ratio of 



Tig. 160. 




the rate of advance to that of rotation is constant, it will expedite matters to operate sys- 
tematically, drawing a number of equidistant vertical lines in the side view and corresponding 
equidistant radii in the end view : a similar system of parallels in the top view will permit the 
projections of these curves to be worked out simultaneously in the three views, which is 
advantageous not only in saving time, but in serving as a check upon the accuracy of the con- 
struction, which is one of considerable delicacy. 

237. In the top view the junction of the body of the blade with the hub is indicated by 



DRAWING OF A SCREW PROPELLER. 141 

sectioning, as in Fig. 127, — just as if the whole blade had been turned off in the lathe, leaving 
a mere film projecting. For the purpose of more clearly indicating the manner in which the 
thickness of the metal is gradually reduced as we recede from the centre and approach the 
thin edge at the periphery, intermediate sections of the blade are also shown in this view. 
These are sections, not by planes, but by cylinders — appearing in the end view as the circles 
fg, hi, kl, and in the side view as the correspondingly lettered helical arcs; in a word, they 
represent successive stages in the supposed process of turning off the blade. The sections of 
the acting face of the blade by these cylinders are therefore true -helices, an aid to drawing 
which in the top view will be found in first determining the tangents at o. A convenient 
method of doing this is to draw at o in the end view a tangent to the periphery, and upon it 
to rectify a convenient fraction oS of the circumference, thus determining oT: in the present 
case the fraction selected was one twelfth. Then in the top view draw a parallel to QQ, at a 
distance from it equal to the same fraction of the pitch ; to this line T is projected from the 
view below, and To thus determined is tangent to the outer helix. In the end view draw 
CT, and intersect it by tangents to the other circles, y^, etc., by which the sections are made: 
the points thus found, projected to T' T' in the top view, will be like manner lie -upon the tan- 
gents to the other helices. 

238. These cylindrical sections then are bounded in front, that is to say, on the acting 
face, by helices. The lines bounding thern^on the back cannot be determined by any fixed 
rule : they are in fact commonly drawn arbitrarily to a great extent, the leading edge being 
always, however, made thinner and sharper than the trailing edge, because in the main the 
propeller is used for going ahead and not for backing. But as a further guide to the distribu- 
tion of the metal, it is a common practice to put in the end view, as shown in full lines, what 
is called a " conventional section." This is not a true section by the plane QQ, however, 
which would give to the back of the blade a form like that shown in the dotted line : the con- 
ventional section indicates the required normal thickness of the metal at points upon the 
vertical element 00. From this we may derive a little assistance in constructing the back 
lines of the cylindrical sections ; thus, for instance, the circle kl cuts 00 at m, and the line of 
the conventional section at « ; in the top view set off On equal to mn, in a direction normal 
to the helix /i9^: then n is one point in the corresponding back line Irnk oi this particular 
section, and a like determination is to be made for each of the others. 

239. In the end view the contour of the leading edge is the curve khf, beyond which 
nothing is visible. The outline of the trailing edge is the curve aligy ; but the thickness of 
the blade at its junction with the hub and for some distance outward is so great that the back, 
it is obvious, can be seen beyond the trailing edge. And the cylindrical sections will enable 
us to ascertain its contour: for example, draw parallel to the axis a tangent to the curve k7il\ 
this line is an element of the cylinder, and touches the curve at some point r, which is 
projected to the circle knl in the end view, thus locating a point in the required visible outline 
of the back ; as many more as desired may be found in like manner. 

240. Since the back lines of the cylindrical sections are, substantially, assumed, it must 
be admitted that there is no absolute certainty that the surface of the back will be " fair," 
that is, free from humps or hollows. To make certain of that, it would be necessary to make 
such sections quite close together, and also a series of sections by transverse planes, 



142 



MECHANICAL DRAWING. 



also close together, and to alter the lines until the two sets were not only concord- 
ant, but composed of fair lines throughout. This, however, it is needless to do unless a 
pattern is to be made in exact accordance with the drawing. Such propellers are usually 
made by "striking up " the face in loam ; this done, a sort of pattern in sand is constructed 
upon it, the back of which the moulders fashion with their trowels : this is used to make a 
mould for the back of the blade, and then destroyed. 

When this course is pursued, a drawing such as has been described serves every purpose. 

241. The appearance of the complete propeller with four blades is shown in Fig. 161. In 




relation to this it may be said that the placing of the two views at such a distance apart as to 
include between their vertical centre-lines just one half the pitch, is rather fanciful perhaps. 
Still, when the proportions are such as to make it convenient, the opportunity thus afforded 
to represent in an appropriate position one half of the outer helix may well be utilized to 
convey graphically an idea of the proportion between the pitch and the diameter of the 
screw, — which can hardly be done in a more effective manner than that here adopted. 

242. In conclusion, a few words may be said in regard to certain modes of sectioning. 
It is by some laid down as an absolute rule that the section-lines should be inclined to the 
outlines at an angle of 45°. But if this be adhered to in the case of a long and thin piece, 
the effect, as any one can satisfy himself by trial, will be very unsatisfactory; an angle of 30° 
with the longer side will produce a much more pleasing effect, and will, besides, require fewer 
lines and consequently less time. 

In the representation of work composed of rolled plates, such as boilers, the details of 



STANDARD SECTIONING, ETC. 143 

iron ships, and the like, the most effective as well as the most expeditious method is to rule 
the sections with fine longitudinal lines, as shown in Fig. 162. 

But if such work is to be drawn upon a small scale, it will often happen that the attempt 
to represent the thickness of a sheet of metal by even a double line is not only laborious, but 
unsuccessful. In such cases a very telling and comparatively easy expedient is shown in the 
lower part of 162 : the whole area of the sectioned parts is filled in with solid black, care being 
taken to show the joinings by leaving a fine line of pure daylight between the surfaces which 
are actually in contact. Fig. 163, which exhibits the side armor of an iron-clad vessel of the 
Monitor type, is an example of the effectiveness of this style for certain purposes in producing 
emphatic contrasts and making conspicuous the important features. But, obviously, contrasts 
so violent as this are suitable only for drawings made on comparatively small scales. 




243. It was stated in (15), that in sectioning the aim should in general be to imitate the 
effect of an even tint covering the surface, so transparent as not to obscure the dotted 
outlines of parts beyond, and therefore of necessity too light to impair the distinctness of the 
lines definino- its own boundaries. It is not imperative that the spacing should be the same 
throughout : if any pieces are small they may be, and if quite small they must be, sectioned 
more closely than the larger ones, though with lines a trifle finer; nor will this make 
them unduly prominent, nor destroy the general effect of uniformity in the tone. This of 
course has reference to the finish only, and is upon the supposition that the ultimate object is 
to produce the most agreeable effect ; the fact that certain portions are cut is distinctly shown, 
but no information as to the materials of the structure is conveyed. 

That information is of course essential to the completeness of a working drawing, and it 
may be conveyed in more than one way. In the days when, before the invention of the blue- 
print process, tracings were sent into the shop, one mode of indicating the different substances 
was by the use of colors, wrought-iron for instance being sectioned in blue, cast-iron in black. 



144 



MECHANICAL DRAWING. 



composition metal in red, and so on. This had the advantage of making the distinctions 
clear and conspicuous, without impairing the even and finished effect. 

244. Colors being inadmissible when blue-printing or kindred processes are employed, the 
practice has obtained to some extent of sectioning different materials in different styles, with 
an effect upon the appearance of the drawing as a whole which in most cases it would be 
gross flattery to call hideous. An effort has been made to secure general adoption of some 
uniform system or standard in regard to this, but as yet without positive results. In Fig. 164 
we give a few specimens selected from the series arranged by Mr. Frank Van Vleck in 




riff.163. 



furtherance of that effort ;* — a series probably as perfect and as free from objections as any 
that could be devised. The materials are as follows : 



A. Wrought-iron. 

B. Steel. 

C. Wires. 

D. Stone. 

E. Brass. 



F. Brick. 

G. Cast-iron. 
H. Wood. 

I. Babbitt-metal. 
K. Vulcanite. 



Aside from the ruinous effect of this method upon the appearance of the drawing, which 
may be regarded by many as of slight importance, it appears open to one or two serious 
practical objections. It is not easy to do the sectioning well in such styles, for instances, as 
those shown in A, B, E, and K, and it is impossible to do it rapidly, which is in many cases of 
considerable importance ; again, absolute dependence upon any system would demand perfect 
familiarity with the adopted code of symbols on the part of all the workmen for whose 
guidance the drawings are made. 



See y4iiiciican Machinist, April 9, 1SS7. 



STANDARD SECTIONING, ETC. 



145 



245. This last would apply with equal force to the use of colors, and indeed it appears in 
any case advisable to make use of reference-letters, of which an explanation should be written 
upon each sheet, as, for example, 

a, a, Cast-iron : b, b, Wrought-iron ; c, c. Steel, etc. ; 

and then let each piece be marked with the proper letter, according to the material of which 
it is to be made. This course, which is both simple and certain, was extensively used even 

















X-.-x- 


-".-':a,^. 


~^\~-^^ 


■ •, '. ' ■■■.-.■.■.■.■.■.. .'.--1 








■ ■ ■ ' : 


'-, ■ ""-X 




■ , ■ ■ 


—■—-.- 


■ ■ 






^^v^ .- 


■z , . 




?.■'•■■■ ■. . ■ ,■. .' -.-1 


t X . . 










-v'-"^'-"^.- 








-'- — 






— -.^-'-^— -: 


5=-—^ 






>.s^ 





^M 



WW/M/A 



% 



^^aZ/x6l 



^//A 






when the sectioning was done in colors : it is equally applicable if the whole drawing is made 
in plain black, thus enabling the draughtsman to avoid the unpleasantly variegated effect of 
the proposed standard sectioning, and to produce results rivaUing in harmony of tone those 
of a well-executed engraving. 



INDEX. 



A 

PAGE 

Accuracv, importance of 3 

Angle, bisecting of 13 

Arcs, approximating 32 

" rectifying of 26 

Arch, Gothic 19, 20 

" parabolic 34, 35 

Archimedean spiral 45, 46 

Arrangement of dranings 15, 16, 55, 5S 

B 

Bevel-wheel, drawing of 133,134 

" " in inclined position I34.- i35 

Border lines 8. 9 

Broad lines 14 

C 

Cavalier projection 126-129 

Centre-lines 57, 59 

Checking accuracy of work 13 

Cissoid 50 

Close-fitting tangent-screw 135 

Companion to the involute 49 

Connecting-rod, stub-end loi, 102 

Contour, visible loi 

Converging lines g 

Copying bj' impression, or transferring 32, S2 

Covering of hemisphere 103 

Cycloid 39 

D 

Development, approximate 102, 103 

Developable helicoid 96-98 

Dotted lines ... n 

E 

Elevations, defined =5 

" sectional ;S 

Ellipses 27-32 

Epicycloid 39, 40 

Epicycloidal teeth 41-43 

Epitrochoid ^_ • - 

Equable or Archimedean spiral 45. 46 

Equiangular or logarithmic spiral 4-^ 4S 



F 

PAGS 

Foreshortening 59, 60 

G 

Gardener's ellipse 28, 29 

Gothic arch 10. 20 

Geometrical progression 4S, 49 

H 

Helix 79 

' ' developed 80 

' ' obliquity of 82 

Hemisphere, covering of 103 

Hyperbola 35~37 

H y pocy cloid 40 

I 

Impression, copying by 32, S2 

Inking in 5 . 11 

Interlaced work 18 

Intersection of cylinders 104-ic^ 

" " and cones loS-iio 

" " " hemisphere rii 

" cones 113,114 

" surfaces of revolution 112 

" screw-propeller blade and hub 115 

Involute 43, 44 

" companion to 49 

Isometric drawing nj 

" projection ny 

" circle, drawing of i n, 

graduating 120 

" angles 121 

" offsets 122 

" scale ii.q 

J 
Junction of lines zs 

L 
Logarithmic spiral 4S. 49 

M 

Mean proportional 48 40 

ia7 



148 



INDEX. 



N 

PAGE 

Newton's square 50,51 

Nut, square-threaded, single 84 

double 85 

" V-threaded 88 

O 

Oblique projections 55, 127 

Orthographic projections 53 

P 

Parabola 33-35 

Parabolic arch 35 

Pencil, use of 2 

Perspective, defined 53 

" pseudo 130 

Prepared ink, use of 7 

Plotting of curves , 49-51 

Progression, geometrical 48, 49 

Projection, abbreviated method of 69-73 

" of circle, abbreviated method of 76 

" cavalier 126-129 

" isometric 117 

" oblique 54,127 

" orthographic 53 

" arrangement of 55 

Projecting lines 53 

Propeller, construction of 139-142 

" complete drawing of 142 

R 

Rectification of arc 26 

Rolled curves or roulettes 37. 38 

Rolling contact 38 

S 

Screw, linear 8l 

" square-threaded, single 81-84 

" double 84, 85 



PAGE 

Screw, Y-threaded, full sharp 86-89 

" " flattened 89 

Screw-propeller, construction of 139-142 

" " complete drawing of. 142 

Sectioning 6, 7 

" of hollow figures , 11 

" of rolled metal 143 

' ' standard 144, 145 

Sectional elevations 58 

Shadow-lines t^ 8 

" cavalier , 1 29 

" isometric ng 

Spiral, Archimedean 45,46 

" logarithmic iig 

" pseudo 47 

Spring, helical, square 85 

" " round 86 

Spur-wheel 41-43 

" " inclined 132,133 

T 

Tangency 13,14,25 

Tangent-screw 135 

Teeth of spur-vsrheel 41-43 

Template for teeth 43 

" " screw 83 

Thickness of lines 5, 14 

Third proportional 48 

Trial and error 24 

Transferring, copying by 32, 82 

V 

Visible contour, line of loi 

Visual rays ., 53 

W 

Worm and wheel, construction 135-138 

Worm-wheel, side view of 138, 139 



